The formula for the derivative of the sum of two functions u v. Derivative of the algebraic sum of functions

due to which equality (3.10) plays an important role both in theoretical studies, as well as in approximate calculations.

The operations of finding the derivative and differential of a function are called differentiation this function. Common name both operations is explained by their obvious dependence. By virtue of formula (3.8), the differential of a function is obtained by simply multiplying its product

relative errors that arise when the increment of a function is replaced by its differential.

Let's find the increment and differential of the function

y = 3(x+ x) 2 + (x+ x) − 3 x2 − x= 6 x x+ 3(x) 2 + x= (6 x+ 1) x+ (x) 2 .

Then dy = (6 x + 1) x . Calculate y and dy at the point x = 1 if x = 0 , 1 y = 7 0 , 1 + 3 0 , 01 = 0 , 73 ; dy = 7 0 , 1 = 0 , 7 .

The absolute error is y − dy = 0.73 − 0.7 = 0.03 and the relative error

y = 0 0 . . 03 73 ≈ 0.04 .

3.5. Derivative of sum, product and quotient functions

Recall the well-known from the course high school differentiation rules, which allow in some cases to find derivatives of functions without resorting directly to the definition.

Theorem 3.3. If the functions u = u (x) and v = v (x)

differentiable

Multiplying these equalities term by term by dx , we get the same rules written in terms of differentials

Proof. Since the proof is carried out in a completely uniform way for all parts of the theorem, we will prove one of them, for example, the second.

Denote y = uv . Let's increment x , and let

is differentiable, and hence continuous (Theorem 3.2), hence

the body can be taken out of the signs of the derivative and differential.

For the case of three factors, successively applying the formula

(3.12), we find

(uvw) ′ = ((uv) w) ′ = (uv) ′ w+ (uv) w′+ (u′ v+ uv′ ) w+ uvw′ = = u ′ vw + uv ′ w + uvw ′.

A similar rule is valid when differentiating the product of any number of factors.

In the following paragraphs, derivatives of the main elementary functions.

3.6. Derivatives of trigonometric functions

Find derivatives of trigonometric functions, namely

Let's get the first one. The increment of the function y \u003d sin x at the point x, with-

The second formula is proved similarly. The third and fourth formulas are obtained if the tangent and cotangent are expressed in terms of sine and cosine and formula (3.13) is used.

3.7. Differentiation of logarithmic functions

Let's prove the first of them. The increment of the function y = log a x at the point x ,

(we used here the identity log a A = log a e ln A ).

3.8. Differentiation complex function.

Derivatives of power and exponential functions

Let the complex function y of the argument x be given by the formulas y = f (u) ,

u = ϕ (x ) (see paragraph 1.4.3)

Theorem 3.4 (on the derivative of a complex function). If functions

increment y of the function y = f (u ) . Since the function y \u003d f (u) is differentiable at the considered point u by the hypothesis of the theorem, then its increment at this point can be represented as (see Definition 3.4)

x → 0 , we arrive at (3.18).

Multiplying equality (3.18) term by term by dx , we obtain an expression for the differential of a complex function

dy = f′ (u)du.

Comment. The differential of the function y \u003d f (u) would have exactly the same form if the argument u were not a function, but an independent variable. This is the so-called invariance property(independence) of the form of the differential with respect to the argument. It should be borne in mind that if u is an independent variable, then du \u003d u is its arbitrary increment, if u is an intermediate argument (that is, a function), then du is the differential of this function, that is, a value that does not coincide with its increment u.

With the help of the last theorem, it is easy to obtain differential formulas

the left side of the last equality is a complex function of x . Differentiating both sides of the last equality with respect to x (the left side as a complex function), we obtain

1 y y ′ =a 1 x ,

y ′ =ay x =ax x a =ax a − 1 .

It is easy to show that this result is also true for x< 0 , если только при

this x α makes sense. Previously, a result was obtained for the case α = n . The second formula is obtained similarly, from which the last formula follows in the particular case for a = e.

Comment. The method of preliminary logarithm, which was used in obtaining the formula for differentiating a power function, has an independent meaning and is called in conjunction with the subsequent finding of the derivative of the logarithm of the function

lnx ) "= cosx lnx + sin x x .

Hence,

y ′ \u003d x sin x (cos x lnx + sin x x)

Comment. The rule of differentiation of a complex function can also be applied to find the derivative of a function given implicitly.

Indeed, if the relationship between x and y is given in the form F ( x , y ) = 0 and this equation is solvable with respect to y , then the derivative y ′ can be found from the equation

3.9. Differentiation of the inverse function.

Differentiation of inverse trigonometric functions

Let two mutually inverse functions y \u003d f (x) and x \u003d ϕ (y)

(see clause 1.4.8).

Theorem 3.5 (on the derivative of the inverse function). If functions

First level

Function derivative. Comprehensive guide (2019)

Imagine a straight road passing through a hilly area. That is, it goes up and down, but does not turn right or left. If the axis is directed horizontally along the road, and vertically, then the road line will be very similar to the graph of some continuous function:

The axis is a certain level of zero height, in life we ​​use sea level as it.

Moving forward along such a road, we are also moving up or down. We can also say: when the argument changes (moving along the abscissa axis), the value of the function changes (moving along the ordinate axis). Now let's think about how to determine the "steepness" of our road? What could this value be? Very simple: how much will the height change when moving forward a certain distance. Indeed, on different sections of the road, moving forward (along the abscissa) one kilometer, we will rise or fall a different number of meters relative to sea level (along the ordinate).

We denote progress forward (read "delta x").

The Greek letter (delta) is commonly used in mathematics as a prefix meaning "change". That is - this is a change in magnitude, - a change; then what is it? That's right, a change in size.

Important: the expression is a single entity, one variable. You should never tear off the "delta" from the "x" or any other letter! That is, for example, .

So, we have moved forward, horizontally, on. If we compare the line of the road with the graph of a function, then how do we denote the rise? Certainly, . That is, when moving forward on we rise higher on.

It is easy to calculate the value: if at the beginning we were at a height, and after moving we were at a height, then. If the end point turned out to be lower than the start point, it will be negative - this means that we are not ascending, but descending.

Back to "steepness": this is a value that indicates how much (steeply) the height increases when moving forward per unit distance:

Suppose that on some section of the path, when advancing by km, the road rises up by km. Then the steepness in this place is equal. And if the road, when advancing by m, sank by km? Then the slope is equal.

Now consider the top of a hill. If you take the beginning of the section half a kilometer to the top, and the end - half a kilometer after it, you can see that the height is almost the same.

That is, according to our logic, it turns out that the slope here is almost equal to zero, which is clearly not true. A lot can change just a few miles away. Smaller areas need to be considered for a more adequate and accurate estimate of the steepness. For example, if you measure the change in height when moving one meter, the result will be much more accurate. But even this accuracy may not be enough for us - after all, if there is a pole in the middle of the road, we can simply slip through it. What distance should we choose then? Centimeter? Millimeter? Less is better!

AT real life measuring the distance to the nearest millimeter is more than enough. But mathematicians always strive for perfection. Therefore, the concept was infinitesimal, that is, the modulo value is less than any number that we can name. For example, you say: one trillionth! How much less? And you divide this number by - and it will be even less. Etc. If we want to write that the value is infinitely small, we write like this: (we read “x tends to zero”). It is very important to understand that this number is not equal to zero! But very close to it. This means that it can be divided into.

The concept opposite to infinitely small is infinitely large (). You've probably already encountered it when you were working on inequalities: this number is greater in modulus than any number you can think of. If you come up with the largest possible number, just multiply it by two and you get even more. And infinity is even more than what happens. In fact, infinitely large and infinitely small are inverse to each other, that is, at, and vice versa: at.

Now back to our road. The ideally calculated slope is the slope calculated for an infinitely small segment of the path, that is:

I note that with an infinitely small displacement, the change in height will also be infinitely small. But let me remind you that infinitely small does not mean equal to zero. If you divide infinitesimal numbers by each other, you can get quite common number, For example, . That is, one small value can be exactly twice as large as another.

Why all this? The road, the steepness ... We are not going on a rally, but we are learning mathematics. And in mathematics everything is exactly the same, only called differently.

The concept of a derivative

The derivative of a function is the ratio of the increment of the function to the increment of the argument at an infinitesimal increment of the argument.

Increment in mathematics is called change. How much the argument () has changed when moving along the axis is called argument increment and denoted by How much the function (height) has changed when moving forward along the axis by a distance is called function increment and is marked.

So, the derivative of a function is the relation to when. We denote the derivative with the same letter as the function, only with a stroke from the top right: or simply. So, let's write the derivative formula using these notations:

As in the analogy with the road, here, when the function increases, the derivative is positive, and when it decreases, it is negative.

But is the derivative equal to zero? Certainly. For example, if we are driving on a flat horizontal road, the steepness is zero. Indeed, the height does not change at all. So with the derivative: the derivative of a constant function (constant) is equal to zero:

since the increment of such a function is zero for any.

Let's take the hilltop example. It turned out that it was possible to arrange the ends of the segment on opposite sides of the vertex in such a way that the height at the ends turns out to be the same, that is, the segment is parallel to the axis:

But large segments are a sign of inaccurate measurement. We will raise our segment up parallel to itself, then its length will decrease.

In the end, when we are infinitely close to the top, the length of the segment will become infinitely small. But at the same time, it remained parallel to the axis, that is, the height difference at its ends is equal to zero (does not tend, but is equal to). So the derivative

This can be understood as follows: when we are standing at the very top, a small shift to the left or right changes our height negligibly.

There is also a purely algebraic explanation: to the left of the top, the function increases, and to the right, it decreases. As we have already found out earlier, when the function increases, the derivative is positive, and when it decreases, it is negative. But it changes smoothly, without jumps (because the road does not change its slope sharply anywhere). Therefore, there must be between negative and positive values. It will be where the function neither increases nor decreases - at the vertex point.

The same is true for the valley (the area where the function decreases on the left and increases on the right):

A little more about increments.

So we change the argument to a value. We change from what value? What has he (argument) now become? We can choose any point, and now we will dance from it.

Consider a point with a coordinate. The value of the function in it is equal. Then we do the same increment: increase the coordinate by. What is the argument now? Very easy: . What is the value of the function now? Where the argument goes, the function goes there: . What about function increment? Nothing new: this is still the amount by which the function has changed:

Practice finding increments:

1. Find the increment of the function at a point with an increment of the argument equal to.
2. The same for a function at a point.

Solutions:

AT different points with the same increment of the argument, the increment of the function will be different. This means that the derivative at each point has its own (we discussed this at the very beginning - the steepness of the road at different points is different). Therefore, when we write a derivative, we must indicate at what point:

Power function.

A power function is called a function where the argument is to some extent (logical, right?).

And - to any extent: .

The simplest case is when the exponent is:

Let's find its derivative at a point. Remember the definition of a derivative:

So the argument changes from to. What is the function increment?

Increment is. But the function at any point is equal to its argument. So:

The derivative is:

The derivative of is:

b) Now consider quadratic function (): .

Now let's remember that. This means that the value of the increment can be neglected, since it is infinitely small, and therefore insignificant against the background of another term:

So, we have another rule:

c) We continue the logical series: .

This expression can be simplified in different ways: open the first bracket using the formula for abbreviated multiplication of the cube of the sum, or decompose the entire expression into factors using the formula for the difference of cubes. Try to do it yourself in any of the suggested ways.

So, I got the following:

And let's remember that again. This means that we can neglect all terms containing:

We get: .

d) Similar rules can be obtained for large powers:

e) It turns out that this rule can be generalized for a power function with an arbitrary exponent, not even an integer:

You can formulate the rule with the words: “the degree is brought forward as a coefficient, and then decreases by”.

We will prove this rule later (almost at the very end). Now let's look at a few examples. Find the derivative of functions:

1. (in two ways: by the formula and using the definition of the derivative - by counting the increment of the function);

1. . Believe it or not, this is a power function. If you have questions like “How is it? And where is the degree? ”, Remember the topic“ ”!
   Yes, yes, the root is also a degree, only a fractional one:.
   So our square root is just a power with an exponent:
   .
   We are looking for the derivative using the recently learned formula:

   If at this point it became unclear again, repeat the topic "" !!! (about the degree with negative indicator)

2. . Now the exponent:

   And now through the definition (have you forgotten yet?):
   ;
   .
   Now, as usual, we neglect the term containing:
   .

3. . Combination of previous cases: .

trigonometric functions.

Here we will use one fact from higher mathematics:

When expression.

You will learn the proof in the first year of the institute (and to get there, you need to pass the exam well). Now I'll just show it graphically:

We see that when the function does not exist - the point on the graph is punctured. But the closer to the value, the closer the function is to. This is the very “strives”.

Additionally, you can check this rule with a calculator. Yes, yes, do not be shy, take a calculator, we are not at the exam yet.

So let's try: ;

Don't forget to switch the calculator to Radians mode!

etc. We see that the smaller, the closer the value of the ratio to.

a) Consider a function. As usual, we find its increment:

Let's turn the difference of sines into a product. To do this, we use the formula (remember the topic ""):.

Now the derivative:

Let's make a substitution: . Then, for infinitely small, it is also infinitely small: . The expression for takes the form:

And now we remember that with the expression. And also, what if an infinitely small value can be neglected in the sum (that is, at).

So we get the following rule: the derivative of the sine is equal to the cosine:

These are basic (“table”) derivatives. Here they are in one list:

Later we will add a few more to them, but these are the most important, as they are used most often.

Practice:

1. Find the derivative of a function at a point;
2. Find the derivative of the function.

Solutions:

1. First we find the derivative in general view, and then substitute its value for it:
   ;
   .
2. Here we have something similar to power function. Let's try to bring her to
   normal view:
   .
   Ok, now you can use the formula:
   .
   .
3. . Eeeeeee….. What is it????

Okay, you're right, we still don't know how to find such derivatives. Here we have a combination of several types of functions. To work with them, you need to learn a few more rules:

Exponent and natural logarithm.

There is such a function in mathematics, the derivative of which for any is equal to the value of the function itself for the same. It is called "exponent", and is an exponential function

The base of this function is a constant - it is infinite decimal, that is, an irrational number (such as). It is called the "Euler number", which is why it is denoted by a letter.

So the rule is:

It's very easy to remember.

Well, let's not go far, let's immediately consider inverse function. What is the inverse of the exponential function? Logarithm:

In our case, the base is a number:

Such a logarithm (that is, a logarithm with a base) is called a “natural” one, and we use a special notation for it: we write instead.

What is equal to? Of course, .

The derivative of the natural logarithm is also very simple:

Examples:

1. Find the derivative of the function.
2. What is the derivative of the function?

Answers: Exhibitor and natural logarithm- functions are uniquely simple in terms of the derivative. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.

Differentiation rules

What rules? Another new term, again?!...

Differentiation is the process of finding the derivative.

Only and everything. What is another word for this process? Not proizvodnovanie... The differential of mathematics is called the very increment of the function at. This term comes from the Latin differentia - difference. Here.

When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:

There are 5 rules in total.

The constant is taken out of the sign of the derivative.

If - some constant number (constant), then.

Obviously, this rule also works for the difference: .

Let's prove it. Let, or easier.

Examples.

Find derivatives of functions:

1. at the point;
2. at the point;
3. at the point;
4. at the point.

Solutions:

1. (the derivative is the same at all points, since it's a linear function, remember?);

Derivative of a product

Everything is the same here: we introduce new feature and find its increment:

Derivative:

Examples:

1. Find derivatives of functions and;
2. Find the derivative of a function at a point.

Solutions:

Derivative of exponential function

Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just the exponent (have you forgotten what it is yet?).

So where is some number.

We already know the derivative of the function, so let's try to bring our function to a new base:

For this we use simple rule: . Then:

Well, it worked. Now try to find the derivative, and don't forget that this function is complex.

Happened?

Here, check yourself:

The formula turned out to be very similar to the derivative of the exponent: as it was, it remains, only a factor appeared, which is just a number, but not a variable.

Examples:
Find derivatives of functions:

Answers:

This is just a number that cannot be calculated without a calculator, that is, it cannot be written in a simpler form. Therefore, in the answer it is left in this form.

Derivative of a logarithmic function

Here it is similar: you already know the derivative of the natural logarithm:

Therefore, to find an arbitrary from the logarithm with a different base, for example, :

We need to bring this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:

Only now instead of we will write:

The denominator turned out to be just a constant (a constant number, without a variable). The derivative is very simple:

Derivatives of the exponential and logarithmic functions are almost never found in the exam, but it will not be superfluous to know them.

Derivative of a complex function.

What is a "complex function"? No, this is not a logarithm, and not an arc tangent. These functions can be difficult to understand (although if the logarithm seems difficult to you, read the topic "Logarithms" and everything will work out), but in terms of mathematics, the word "complex" does not mean "difficult".

Imagine a small conveyor: two people are sitting and doing some actions with some objects. For example, the first wraps a chocolate bar in a wrapper, and the second ties it with a ribbon. It turns out such a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the opposite in reverse order.

Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then we will square the resulting number. So, they give us a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, in order to find its value, we do the first action directly with the variable, and then another second action with what happened as a result of the first.

We may well do the same actions in reverse order: first you square, and then I look for the cosine of the resulting number:. It is easy to guess that the result will almost always be different. Important feature complex functions: when you change the order of actions, the function changes.

In other words, A complex function is a function whose argument is another function: .

For the first example, .

Second example: (same). .

The last action we do will be called "external" function, and the action performed first - respectively "internal" function(these are informal names, I use them only to explain the material in simple language).

Try to determine for yourself which function is external and which is internal:

Answers: The separation of inner and outer functions is very similar to changing variables: for example, in the function

1. What action will we take first? First we calculate the sine, and only then we raise it to a cube. So it's an internal function, not an external one.
   And the original function is their composition: .
2. Internal: ; external: .
   Examination: .
3. Internal: ; external: .
   Examination: .
4. Internal: ; external: .
   Examination: .
5. Internal: ; external: .
   Examination: .

we change variables and get a function.

Well, now we will extract our chocolate - look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. For the original example, it looks like this:

Another example:

So, let's finally formulate the official rule:

Algorithm for finding the derivative of a complex function:

Everything seems to be simple, right?

Let's check with examples:

Solutions:

1) Internal: ;

External: ;

2) Internal: ;

(just don’t try to reduce by now! Nothing is taken out from under the cosine, remember?)

3) Internal: ;

External: ;

It is immediately clear that there is a three-level complex function here: after all, this is already a complex function in itself, and we still extract the root from it, that is, we perform the third action (put chocolate in a wrapper and with a ribbon in a briefcase). But there is no reason to be afraid: anyway, we will “unpack” this function in the same order as usual: from the end.

That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.

In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:

The later the action is performed, the more "external" the corresponding function will be. The sequence of actions - as before:

Here the nesting is generally 4-level. Let's determine the course of action.

1. Radical expression. .

2. Root. .

3. Sinus. .

4. Square. .

5. Putting it all together:

DERIVATIVE. BRIEFLY ABOUT THE MAIN

Function derivative- the ratio of the increment of the function to the increment of the argument with an infinitesimal increment of the argument:

Basic derivatives:

Differentiation rules:

The constant is taken out of the sign of the derivative:

Derivative of sum:

Derivative product:

Derivative of the quotient:

Derivative of a complex function:

Algorithm for finding the derivative of a complex function:

1. We define the "internal" function, find its derivative.
2. We define the "external" function, find its derivative.
3. We multiply the results of the first and second points.

Questions for the exam in the academic discipline "Elements of Higher Mathematics"

for specialty 230115 "Programming in computer systems"

2012\2013 academic year.

Matrices and actions on them.

(O. A null matrix is ​​a matrix with all elements equal to 0.

O. Two matrices of the same dimension mxn are called equal if at the intersection i-th line and the j-th column in one and the other matrix contains the same number; i=1, 2, ..., m ; j=1, 2, ..., n .

Let be A= (a ij) is some matrix and g is an arbitrary number, then g A= (g a ij), that is, when multiplying the matrix A by the number g, all the numbers that make up the matrix A are multiplied by the number g.

Let A and B be matrices of the same dimension A = (a ij), B = (b ij), then their sum A + B is a matrix C = (c ij) of the same dimension, determined from the formula c ij = a ij + b ij , that is, when adding two matrices, the numbers equally located in them are added in pairs.

Matrix A can be multiplied by matrix B, that is, find matrix C = AB, if the number of columns n of matrix A is equal to the number of rows of matrix B, while matrix C will have as many rows as there are rows in matrix A and as many columns as there are columns in matrix B. Each element of the matrix C is defined by a formula.

Element c ij of matrix-product C is equal to the sum products of the elements of the i -row of the first matrix - factor by the corresponding elements of the j -th column of the second matrix - factor.

The concept of a determinant and its properties.

This term has other meanings as well. Determinant (disambiguation) .

Determinant(or determinant) is one of the basic concepts linear algebra. Determinant matrices is an polynomial from the elements of a square matrix (that is, one in which the number of rows and columns is equal). In general matrix can be defined over any commutative ring, in which case the determinant will be an element of the same ring.

PROPERTY 1. The value of the determinant will not change if all its rows are replaced by columns, and each row is replaced by a column with the same number, that is

PROPERTY 2. Permuting two columns or two rows of a determinant is equivalent to multiplying it by -1.

PROPERTY 3. If a determinant has two identical columns or two identical rows, then it zero.

PROPERTY 4. Multiplying all elements of one column or one row of a determinant by any number k is equivalent to multiplying the determinant by this number k.

PROPERTY 5. If all elements of some column or some row are equal to zero, then the determinant itself is equal to zero. This property is a special case of the previous one (for k=0).

PROPERTY 6. If the corresponding elements of two columns or two rows of a determinant are proportional, then the determinant is equal to zero.

PROPERTY 7. If each element of the nth column or nth row of the determinant is the sum of two terms, then the determinant can be represented as the sum of two determinants, of which one in the nth column or, respectively, in the nth row has the first from the mentioned terms, and the other - the second; the elements in the remaining places are the same for the milestones of the three determinants.

PROPERTY 8. If we add to the elements of some column (or some row) the corresponding elements of another column (or another row), multiplied by any common factor, then the value of the determinant will not change. For example. Further properties of determinants are connected with the concept of algebraic complement and minor. The minor of some element is the determinant obtained from the given one by deleting the row and column at the intersection of which this element is located.

The algebraic complement of any element of the determinant is equal to the minor of this element, taken with its sign, if the sum of the row and column numbers at the intersection of which the element is located is an even number, and with the opposite sign if this number is odd.

We will denote the algebraic complement of an element by a capital letter of the same name and the same number as the letter that denotes the element itself.

PROPERTY 9. The determinant is equal to the sum of the products of the elements of any column (or row) and their algebraic complements. In other words, the following equalities hold:

Calculation of determinants.

The calculation of determinants is based on their known properties, which apply to determinants of all orders. These properties are:

1. If you rearrange two rows (or two columns) of the determinant, then the determinant will change sign.

2. If the corresponding elements of two columns (or two rows) of the determinant are equal or proportional, then the determinant is equal to zero.

3. The value of the determinant will not change if the rows and columns are swapped, preserving their order.

4. If all elements of any row (or column) have a common factor, then it can be taken out of the determinant sign.

5. The value of the determinant will not change if the corresponding elements of another row (or column) are added to the elements of one row (or column), multiplied by the same number. For third-order determinants, this property can be written, for example, as follows:

6. The second order determinant is calculated by the formula

7. The third order determinant is calculated by the formula

There is a convenient scheme for calculating the third order determinant (see Fig. 1 and Fig. 2).

According to the scheme shown in fig. 1, the products of the connected elements are taken with their own sign, and according to the scheme of Fig. 2 - with the opposite. The value of the determinant is equal to the algebraic sum of the six products obtained.

Systems of linear equations. Basic concepts and definitions.

Systemmr linear algebraic equations withn unknown(or, linear system, also used abbreviation SLAU) in linear algebra is a system of equations of the form

System linear equations from three variables defines a set planes. The point of intersection is the solution.

Here is the number of equations, and is the number of unknowns. x 1 , x 2 , …, x n are unknowns to be determined. a 11 , a 12 , …, a mn- system coefficients - and b 1 , b 2 , … b m- free members - assumed to be known . Coefficient indices ( a ij) systems denote the numbers of the equation ( i) and unknown ( j), at which this coefficient stands, respectively .

System (1) is called homogeneous if all its free terms are equal to zero ( b 1 = b 2 = … = b m= 0), otherwise - heterogeneous.

System (1) is called square if the number m equations is equal to the number n unknown.

Decision systems (1) - set n numbers c 1 , c 2 , …, c n, such that the substitution of each c i instead of x i into system (1) turns all its equations into identities.

System (1) is called joint if it has at least one solution, and incompatible if it has no solution.

A joint system of the form (1) may have one or more solutions.

Solutions c 1 (1) , c 2 (1) , …, c n(1) and c 1 (2) , c 2 (2) , …, c n(2) joint systems of the form (1) are called various if at least one of the equalities is violated:

A joint system of the form (1) is called certain if it has a unique solution; if it has at least two different solutions, then it is called uncertain. If there are more equations than unknowns, it is called redefined .

Methods for solving systems of linear equations (Cramer and Gauss method).

Gauss method - classical solution method linear systems algebraic equations (SLAU). This is a sequential elimination method variables, when, with the help of elementary transformations, the system of equations is reduced to an equivalent system of a triangular form, from which all other variables are found sequentially, starting from the last (by number) variables .

Cramer's method (Cramer's rule)- a way to solve square systems of linear algebraic equations with non-zero determinant main matrix(moreover, for such equations the solution exists and is unique). named after Gabriel Kramer(1704–1752), who invented the method.

Vectors. Linear operations on them.

A directed segment is called a vector. If the beginning of the vector is at point A and the end is at point B, then the vector is denoted AB. If the beginning and end of the vector are not indicated, then it is denoted by a lowercase letter of the Latin alphabet a, b, c, .... BA denotes a vector directed opposite to the vector AB. A vector whose beginning and end coincide is called a null vector and is denoted by ō. Its direction is uncertain.

The length or modulus of a vector is the distance between its beginning and end. Records |AB| and |a| denote the moduli of the vectors AB and a.

Vectors are called collinear if they are parallel to the same line, and coplanar if they are parallel to the same plane.

Two vectors are said to be equal if they are collinear, have the same direction, and are equal in length.

Linear operations on vectors include:

1) multiplication of a vector by a number (The product of a vector a and a number α is a vector denoted by α∙a. (or vice versa a∙α), whose modulus is |α a| =|α||a|, and the direction coincides with the direction of the vector a if α>0 and vice versa if α< 0.

2) addition of vectors (The sum of vectors is a vector, denoted by , whose beginning is at the beginning of the first vector a 1, and the end is at the end of the last vector a n , a polyline composed of a sequence of summand vectors. This addition rule is called the polyline closure rule. In the case the sum of two vectors, it is equivalent to the parallelogram rule)

A straight line e with a direction given on it, taken as positive, is called the e-axis.

A linear combination of vectors a i is a vector a defined by the formula , where are some numbers.

If for a system of n vectors a i the equality

is true only if this system is called linearly independent. If equality (1) holds for , at least one of which is nonzero, then the system of vectors ai is called linearly dependent. For example, any collinear vectors, three coplanar vectors, four or more vectors in three-dimensional space are always linearly dependent.

Three ordered linearly independent vectors ē 1 , ē 2 , ē 3 in space is called a basis. An ordered triple of non-coplanar vectors always forms a basis. Any vector a in space can be expanded in terms of the basis ē 1 , ē 2 , ē 3 , i.e. represent a as a linear combination of basis vectors: a= xē 1 + yē 2 + zē 3 , where x, y, z are the coordinates vector a in the basis ē 1 , ē 2 , ē 3 . A basis is called orthonormal if its vectors are mutually perpendicular and have unit length. Such a basis is denoted by i, j, k, i.e. i=(1, 0, 0), j=(0, 1, 0), k=(0, 0, 1).

Example 5. Vectors are given in the orthonormal basis i, j, k with coordinates: a=(2;-1;8), e 1 = (1,2,3), e 2 = (1,-1,-2), e 3 \u003d (1, -6.0). Make sure that the triple e 1, e 2, e 3 forms a basis, and find the coordinates of the vector in this basis.

Decision. If the determinant , composed of the coordinates of the vectors e 1, e 2, e 3, is not equal to 0, then the vectors e 1, e 2, e 3 are linearly independent and, therefore, form a basis. We make sure that \u003d -18-4 + 3-12 \u003d -31 Thus, the triple e 1, e 2, e 3 is the basis.

Let us denote the coordinates of the vector a in the basis e 1 , e 2 , e 3 through x,y,z. Then a \u003d (x, y, z) \u003d xe 1 + ye 2 + ze 3. Since according to the condition a \u003d 2i - j + 8k, e 1 \u003d i + 2j + 3k, e 2 \u003d i - j -2k, e 3 \u003d i - 6j, then from the equality a \u003d xe1 + ye 2 + ze 3 follows 2i – j +8k = xi + 2xj + 3xk + yi – yj -2yk +zi -6zj = (x+y+z)i +(2x-y-6z)j +(3x-2y)k.. As you can see, the vector on the left side of the resulting equality is equal to the vector on its right side, and this is possible only if their corresponding coordinates are equal. From here we get a system for finding unknowns x, y, z:

Its solution: x = 2, y = -1, z = 1. So, a = 2e 1 - e 2 + e 3 = (2,-1,1).

Decomposition of vectors. Scalar product vectors.

Scalar product sometimes inner product- operation on two vectors, the result of which is the number ( scalar), which does not depend on the coordinate system and characterizes the lengths of the multiplier vectors and the angle between them. This operation corresponds to the multiplication length vector x on projection vector y to vector x. This operation is usually considered as commutative and linear for each factor.

One of the following notations is usually used:

or ( designation Dirac, often used in quantum mechanics for state vectors):

It is usually assumed that the dot product is positive definite, that is,

For all .

If this is not assumed, then the work is called indefinite.

Dot product in vector space above field integrated(or material) numbers is a function for elements that takes values ​​in (or ), defined for each pair of elements and satisfying the following conditions:

Note that it follows from item 2 of the definition that . Therefore, item 3 makes sense, despite the complex (in the general case) values dot product.

Vector product of vectors.

vector product- This pseudovector, perpendicular plane constructed by two factors, which is the result of binary operation"vector multiplication" over vectors in 3D Euclidean space. The product is neither commutative, nor associative(it is anticommutative) and differs from dot product of vectors. In many engineering and physics problems, it is necessary to be able to build a vector perpendicular to two existing ones - the vector product provides this opportunity. The cross product is useful for "measuring" the perpendicularity of vectors - the length of the cross product of two vectors is equal to the product of their lengths if they are perpendicular, and decreases to zero if the vectors are parallel or anti-parallel.

You can define a vector product in different ways, and theoretically, in a space of any dimension n product can be calculated n-1 vectors, thus obtaining a single vector perpendicular to them all. But if the product is limited to non-trivial binary products with vector results, then the traditional vector product is defined only in three-dimensional and seven-dimensional spaces. Result vector product, as well as scalar, depends on metrics Euclidean space.

In contrast to the formula for calculating the coordinates of vectors dot product in 3D rectangular coordinate system, the formula for the cross product depends on orientation rectangular coordinate system or, in other words, its " chirality».

Mixed product of vectors

Mixed product vectors - scalar product vector on the vector product vectors and :

Sometimes it is called triple scalar product vectors, apparently due to the fact that the result is scalar(more precisely - pseudoscalar).

Geometric sense: The modulus of the mixed product is numerically equal to the volume parallelepiped educated vectors .

mixed product skew-symmetric with respect to all its arguments:

i.e., a permutation of any two factors changes the sign of the product. Hence it follows that

In particular,

The mixed product is conveniently written as symbol (tensor) Levi-Civita:

(in the last formula in an orthonormal basis, all indices can be written as lower ones; in this case, this formula repeats the formula with a determinant completely directly, however, this automatically results in a factor (-1) for left bases).

Cartesian rectangular coordinate system on the plane.

Let's take two mutually perpendicular straight lines on the plane - two coordinate axes Ox and Oy with positive directions indicated on them (Fig. 1). The lines Ox and Oy are called coordinate axes, the point of their intersection O is the origin of coordinates.

The coordinate axes Ox, Oy with the selected scale unit are called the Cartesian rectangular (or rectangular) coordinate system on the plane.

We assign two numbers to an arbitrary point M of the plane: the abscissa x, equal to the distance from the point M to the Oy axis, taken with the “+” sign if M lies to the right of Oy, and with the “-” sign if M lies to the left of Oy; the y-ordinate equal to the distance from the point M to the Ox axis, taken with the “+” sign if M lies above Ox, and with the “-” sign if M lies below Ox. The abscissa x and the ordinate y are called the Cartesian rectangular coordinates of the point M(x; y).

The origin of coordinates has coordinates (0;0). The coordinate axes divide the plane into four parts called quarters or quadrants (sometimes also called coordinate angles). The part of the plane enclosed between the positive semi-axes Ox and Oy is called the first quadrant. Further, the numbering of the quadrants goes counterclockwise (Fig. 2). For all points of the I quadrant x>0, y>0; for points I I quadrant x<0, у>0, in I I I quadrant x<0, у<0 и в IV квадранте х>0, y<0.

Polar coordinates.

Polar coordinate system- a two-dimensional coordinate system in which each point on the plane is determined by two numbers - a polar angle and a polar radius. The polar coordinate system is especially useful when relationships between points are easier to represent as radii and angles; in the more common Cartesian or rectangular coordinate system, such relationships can only be established by applying trigonometric equations.

The polar coordinate system is given by a ray, which is called the zero or polar axis. The point from which this ray emerges is called the origin or pole. Any point on the plane is defined by two polar coordinates: radial and angular. The radial coordinate (usually denoted ) corresponds to the distance from the point to the origin. Angular coordinate, also called polar angle or azimuth and denoted , is equal to the angle by which you need to turn the polar axis counterclockwise in order to get to this point.

The radial coordinate defined in this way can take values ​​from zero before infinity, and the angular coordinate varies from 0° to 360°. However, for convenience, the range of values ​​of the polar coordinate can be extended beyond the limit

Equation of a straight line on a plane

Ah + Wu + C = 0,

and the constants A, B are not equal to zero at the same time. This first order equation is called the general equation of a straight line. Depending on the values ​​of the constants A, B and C, the following special cases are possible:

C \u003d 0, A ≠ 0, B ≠ 0 - the line passes through the origin

A \u003d 0, B ≠ 0, C ≠ 0 (By + C \u003d 0) - the line is parallel to the Ox axis

B \u003d 0, A ≠ 0, C ≠ 0 ( Ax + C \u003d 0) - the line is parallel to the Oy axis

B \u003d C \u003d 0, A ≠ 0 - the straight line coincides with the Oy axis

A \u003d C \u003d 0, B ≠ 0 - the straight line coincides with the Ox axis

The equation of a straight line can be presented in various forms depending on any given initial conditions.

The main tasks of using the equation of a straight line

I can not answer

Curves of the second order

Curve of the second order- locus of points whose Cartesian rectangular coordinates satisfy an equation of the form

in which at least one of the coefficients is different from zero.

Number sequence limit and functions

The limit of the numerical sequence. Consider a numerical sequence whose common term approaches a certain number a by increasing the serial number n. In this case, the number sequence is said to have limit. This concept has a more rigorous definition.

This definition means that a there is limit number sequence if its common term indefinitely approaches a with increasing n. Geometrically, this means that for any > 0 one can find such a number N, that starting from n > N all the terms of the sequence are located inside the interval ( a a). A sequence that has a limit is called converging; otherwise - divergent.

The sequence is called limited if there is such a number M what | u n | M  for all n . An ascending or descending sequence is called monotonous.

Basic limit theorems and their application

Theorem 1 . (on passing to the limit in equality) If two functions take the same values ​​in a neighborhood of some point, then their limits at this point coincide.

Theorem 2. (on passing to the limit in inequality) If the function values f(x) in a neighborhood of some point do not exceed the corresponding values ​​of the function g(x) , then the limit of the function f(x) at this point does not exceed the limit of the function g(x) .

Theorem 3 . The limit of a constant is equal to the constant itself.

Proof. f(x)=s, we will prove that .

Take an arbitrary >0. As  you can take any

positive number. Then at

Theorem 4. Function cannot have two different limits in

one point.

Proof. Let's assume the opposite. Let be

and .

By theorem on the connection between the limit and an infinitesimal function:

f(x)- A= - b.m. at ,

f(x)- B= - b.m. at .

Subtracting these equalities, we get:

B-A= - .

Passing to the limits in both parts of the equality for , we have:

B-A=0, i.e. B=A. We obtain a contradiction proving the theorem.

Theorem 5. If each term algebraic sum functions has a limit at , then the algebraic sum has a limit at , and the limit of the algebraic sum is equal to the algebraic sum of the limits.

.

Proof. Let be , , .

Then, by the theorem on the connection of the limit and b.m. functions:

where - b.m. at .

We add these equalities algebraically:

f(x)+ g(x)- h(x)-(A+B-C)= ,

where b.m. at .

According to the theorem on the connection between the limit and the b.m. features:

A+B-C= .

Theorem 6. If each of the factors of the product of a finite number of functions has a limit at , then the product also has a limit at , and the limit of the product is equal to the product of the limits.

.

Consequence. The constant factor can be taken out of the limit sign.

.

Theorem 7. If functions f(x) and g(x) have a limit at ,

and , then their quotient also has a limit at , and the limit of the quotient is equal to the quotient of the limits.

, .

Function continuity

On fig. 15, and the graph of the function is shown . It is natural to call it a continuous graph, because it can be drawn with a single stroke of the pencil without leaving the paper. Let's set an arbitrary point (number). Another point close to it can be written as , where there is a positive or negative number, called increment . Difference

is called the increment of the function at the point corresponding to the increment . What is meant here is that . On fig. 15, as well as the length of the segment.

We will tend to zero; then for the function under consideration, obviously, and will tend to zero:

. (1)

Consider now the graph in Figure 15, b. It consists of two continuous pieces and . However, these pieces are not connected continuously, and therefore it is natural to call the graph discontinuous. In order for the graph to depict a single-valued function at the point, we agree that it is equal to the length of the segment connecting and; as a sign of this, the point is depicted on the graph by a circle, while an arrow is drawn near the point, indicating that it does not belong to the graph. If the point belonged to the graph, then the function would be two-valued at the point.

Let us now give an increment and determine the corresponding increment of the function:

If we tend to zero, then it is no longer possible to say what will tend to zero. For negative tending to zero, this is true, but for positive ones it is not so at all: it can be seen from the figure that if, remaining positive, tends to zero, then the corresponding increment tends to a positive number equal to the length of the segment.

After these considerations, it is natural to call a function given on a segment continuous at a point of this segment if its increment at this point, corresponding to the increment of , tends to zero in any way of tending to zero. This (the property of continuity in ) is written in the form of relation (1) or else like this:

Record (2) reads as follows: the limit is zero when it tends to zero according to any law. However, the expression "according to any law" is usually omitted, implying it.

If a function defined on is not continuous at the point , i.e., if property (2) does not hold for it in at least one way of tending to zero, then it is called discontinuous at the point .

The function shown in fig. 15, a, is continuous at any point , while the function depicted in fig. 15b is obviously continuous at any point , except for the point , because for the latter, relation (2) is not satisfied when , remaining positive.

A function that is continuous at any point of a segment (interval) is called continuous on this segment (interval).

A continuous function mathematically expresses a property that we often encounter in practice, which consists in the fact that a small increment of an independent variable corresponds to a small increment of a variable (function) dependent on it. Excellent examples of a continuous function are various laws of motion of bodies, expressing the dependence of the path traveled by the body on time. Time and space are continuous. One or another law of motion establishes between them a certain continuous connection, characterized by the fact that a small increment of time corresponds to a small increment of the path.

Man came to the abstraction of continuity by observing the so-called continuous media surrounding him - solid, liquid or gaseous, for example, metals, water, air. In fact, any physical environment is an accumulation a large number moving particles separated from each other. However, these particles and the distances between them are so small compared to the volumes of the media that one has to deal with in macroscopic physical phenomena, that many such phenomena can be studied quite well if we assume that the mass of the medium under study is approximately continuously distributed without any gaps in the space occupied by it. Many physical disciplines are based on this assumption, for example, hydrodynamics, aerodynamics, and the theory of elasticity. The mathematical concept of continuity naturally plays a large role in these disciplines, as in many others.

Continuous functions form the main class of functions with which mathematical analysis operates.

Examples continuous functions elementary functions can serve (see § 3.8 below). They are continuous at intervals where they are defined.

Discontinuous functions in mathematics reflect jump processes occurring in nature. On impact, for example, the velocity of a body changes abruptly. Many qualitative transitions are accompanied by jumps. For example, the relationship between the temperature of one gram of water (ice) and the number of calories of heat contained in it, when it changes between and , if we conditionally assume that when the value , is expressed by the following formulas:

We assume that the heat capacity of ice is 0.5. When this function turns out to be indefinite - multivalued; For convenience, we can agree that for it takes a well-defined value, for example, . The function , obviously discontinuous at , is shown in Fig. sixteen.

Let us give a definition of the continuity of a function at a point.

A function is called continuous at a point if it is defined in some neighborhood of this point, including at the point itself, and if its increment at this point, corresponding to the increment of the argument , tends to zero at :

If we put , then we obtain the following equivalent definition of continuity at : a function is continuous at a point if it is defined in some neighborhood of this point, including at the point itself, and if

; (4)

or also in the language , : if for every there is such that

Equality (4) can also be written as follows:

. (4’)

It shows that under the sign of a continuous function it is possible to pass to the limit.

Example 1. A constant is a function that is continuous at any point . Indeed, the point corresponds to the value of the function, the point corresponds to the same value . That's why

.

Example 2. The function is continuous for any value of , because and, therefore, when .

Example 3. The function is continuous for any . Indeed,

But for any there is an inequality

If , then this follows from Fig. 17, where a circle of radius 1 is shown (an arc of length greater than the chord it contracts, which has a length of ). For , inequality (6) becomes an equality. If , then . Finally, if , then . From (5) on the basis of (6) it follows

,

But then obviously

It can also be said that for any one can find , just such that

We note an important theorem.

THEOREM 1. If the functions and are continuous at the point , then their sum, difference, product, and quotient (at ) are also continuous at this point.

This theorem follows directly from Theorem 6 of §3.2, if we take into account that in this case

There is also an important theorem on the continuity of a function of a function (complex function).

THEOREM 2. Let a function be given that is continuous at the point and another function that is continuous at the point , and let . Then the complex function is continuous at the point .

Proof. Note that by the definition of the continuity of a function at a point, it follows that it is defined in some neighborhood of this point. So

Here a substitution is introduced and the continuity at the point is taken into account .

Example 4. Function

where are constant coefficients, is called a polynomial of degree. It is continuous for any . After all, in order to get, it is necessary, based on constant numbers and functions, to perform a finite number of arithmetic operations - addition, subtraction and multiplication. But the constant is a continuous function (see Example 1), and the function is also continuous (see Example 2), so continuity follows from Theorem 1.

EXAMPLE 5. The function is continuous. It is a composition of two continuous functions: , .

Example 6. Function

is continuous for the specified , because (see Theorem 1) it is equal to the quotient of the division of continuous functions and, moreover, the divisor is not equal to zero (for the specified ).

Example 7. Function

is continuous for any , because it is a composition of continuous functions: , , (see Theorem. 2).

Example 8. The function is continuous because

Example 9. If a function is continuous at a point, then the function is also continuous at that point.

This follows from Theorem 2 and Example 8, because a function is a composition of two continuous functions, .

We note two more theorems that follow directly from the corresponding Theorems 1 and 2 of §3.2 for the limit of a function.

THEOREM 3. If a function is continuous at a point , then there is a neighborhood of this point on which it is bounded.

THEOREM 4. If the function is continuous at the point and , then there exists a neighborhood of the point on which

.

Moreover, if , then

and if , then

The concept of a derivative.

Derivative(functions at a point) - basic concept differential calculus characterizing the rate of change of the function (at a given point). Defined as limit the ratio of the increment of a function to its increment argument when trying to increment the argument to zero if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at a given point).

The process of calculating the derivative is called differentiation. Reverse process - finding primitive - integration.

Geometrical and mechanical meaning of the derivative..

Differentiation rules.

Derivative of the algebraic sum of functions

Theorem 1. Derivative the sum (difference) of two differentiable functions is equal to the sum (difference) of the derivatives of these functions:

(u±v)" = u"±v"

Consequence. The derivative of a finite algebraic sum of differentiable functions is equal to the same algebraic sum of derivative terms. For example,

(u - v + w)" = u" - v" + w"

The derivative of the product of functions is defined by

Theorem 2. The derivative of the product of two differentiable functions is equal to the product of the first function and the derivative of the second plus the product of the second function and the derivative of the first, i.e.

(uv)" = u"v + uv"

Corollary 1. The constant factor can be taken out of the sign of the derivative (cv)" = cv" (с = const).

Corollary 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of them and all the others.

For example, (uvw)" = u"vw + uv"w + uvw"

The derivative of the quotient of two functions

is expressed by the following theorem.

Theorem 3. The derivative of the quotient of two differentiable functions is defined by the formula

The derivative of a complex function is expressed by

Theorem 4. If y = f(u) and u = (φ(x)) are differentiable functions of their arguments, then derivative of a compound function y \u003d f (φ (x)) exists and is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable, i.e.

Very often in tests in mathematics for derivatives complex functions are given, for example, y = sin(cos5x). The derivative of such a function is -5sin5x*sin(cos5x)

See an example of calculating a complex function in the following video

Derivatives of elementary functions.

Let us formulate a necessary condition for the existence of a derivative.

Theorem.

If a function is differentiable at some point, then the function is continuous at that point.

Note that the converse is not true: a continuous function may not have a derivative.

For example, the function is continuous for
, but not differentiable for this value, since at the point
function graph
there is no tangent.

Thus, the continuity of a function is a necessary but not sufficient condition for the differentiability of a function.

4.4. Derivative of sum, difference, product and quotient functions

Finding the derivative of a function directly by definition (section 4.1) is often associated with certain difficulties. In practice, functions are differentiated using a number of rules and formulas.

Theorem.

If functions
and
differentiable at a point X, then at this point the functions
,
,(provided that
) and wherein

;

;

,
.

Consequences

1.
, where
.

2. If
, then.

3.
, where
.

4.6. Derivative of a complex function

Let be
and
, then
− complex function with an intermediate argument u and independent argument X.

Theorem.

If functions
has a derivative
at the point X, and the function
has a derivative
at the corresponding point
, then the complex function
at the point X has a derivative
, which is found by the formula:

or
=.

Briefly, this can be stated as chain rule): the derivative of a complex function is equal to the product of the derivatives of the functions of its constituents.

This rule applies to complex functions with any (certain) number of intermediate arguments.

So if
,
,
,
, then

4.7. Derivative of inverse function

If a
and
are mutually inverse differentiable functions and
, then

or
,

those. the derivative of the inverse function is equal to the reciprocal of the derivative of the given function.

Write down:

or .

Example

Find the derivative of a function
.

,
, then
,
. We have
.

.

So,
.

4.8. Derivative table

For convenience and simplification of the differentiation process, the formulas for the derivatives of the basic elementary functions and the differentiation rules are summarized in a table.

4.9. Examples of finding derivatives of complex functions

In practice, one often has to find derivatives of complex functions. Let us show by examples how to find derivatives of such functions.

1.
,k− number.

;

2.
.

;

3.
.

;

4.
.

;

.

5.
.

;

6.
.

;

;

.

7.
.

.

8.
.

9.
.

10.
.

.

For the case of differentiation of complex functions, the table of derivatives can be rewritten in a more general form.

Formulas for differentiating basic elementary functions from an intermediate argument (
)

4.10. Derivative of a function defined parametrically

Dependence between variables X and y can be set parametrically in the form of two equations:

where t− auxiliary variable (parameter).

Function
, defined by these equations, can be considered as a complex function
, where
.

According to the rule of differentiation of a complex function, we have:

.

As
, then

.

Examples

Find derivatives of functions:

1.

.

2.

.

4.11 Derivative of an implicit function

If the implicit function is given by the equation
, then to find the derivative of at on X we need to differentiate this equation with respect to X while considering at as a function of X, and then, solve the resulting equation with respect to , expressing through X and at.

Example

Find the derivative of a function:
.

;

.

4.12. Logarithmic differentiation

In a number of cases, when it is necessary to differentiate the product of many factors or a quotient in which both the numerator and denominator consist of several factors, as well as when finding derivatives of an exponential-power function
, apply logarithmic differentiation.

The method of logarithmic differentiation is that from given function at first, the natural logarithm is found, and then the result is differentiated:

.

From the resulting equality is determined :

.

Examples

Find derivatives of functions:

1.
.

;

;

2.
.

;

;

;

.

4.13. Derivatives of higher orders

Derivative
from function
called derivative of the first order(or the first derivative) and is a function of X.

The derivative of the first derivative is called second order derivative or the second derivative and denote
,
,.

So by definition

.

The second derivative plays the role of accelerating the change in the function.

The derivative of the second order derivative is called third order derivative and denoted
,
,.

Thus,

.

derivative n-th order (or n th derivative) is called the derivative of the derivative ( n-1) order:

.

Number n, indicating the order of the derivative, is enclosed in brackets so as not to be confused with the exponent.

Derivatives of order higher than the first are called derivatives of higher orders.

The order of the derivative, starting from the fourth, is indicated by Roman numerals or Arabic numerals in brackets, for example,
or
etc.

The operation of finding a derivative is called differentiation.

As a result of solving problems of finding derivatives of the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716) were the first to work in the field of finding derivatives.

Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but only need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.

To find the derivative, you need an expression under the stroke sign break down simple functions and determine what actions (product, sum, quotient) these functions are related. Further, we find the derivatives of elementary functions in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient - in the rules of differentiation. The table of derivatives and differentiation rules are given after the first two examples.

Example 1 Find the derivative of a function

Decision. From the rules of differentiation we find out that the derivative of the sum of functions is the sum of derivatives of functions, i.e.

From the table of derivatives, we find out that the derivative of "X" is equal to one, and the derivative of the sine is cosine. We substitute these values ​​in the sum of derivatives and find the derivative required by the condition of the problem:

Example 2 Find the derivative of a function

Decision. Differentiate as a derivative of the sum, in which the second term with a constant factor, it can be taken out of the sign of the derivative:

If there are still questions about where something comes from, they, as a rule, become clear after reading the table of derivatives and the simplest rules of differentiation. We are going to them right now.

Table of derivatives of simple functions

1. Derivative of a constant (number). Any number (1, 2, 5, 200...) that is in the function expression. Always zero. This is very important to remember, as it is required very often
2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember
3. Derivative of degree. When solving problems, you need to convert non-square roots to a power.
4. Derivative of a variable to the power of -1
5. Derivative square root
6. Sine derivative
7. Cosine derivative
8. Tangent derivative
9. Derivative of cotangent
10. Derivative of the arcsine
11. Derivative of arc cosine
12. Derivative of arc tangent
13. Derivative of the inverse tangent
14. Derivative of natural logarithm
15. Derivative of a logarithmic function
16. Derivative of the exponent
17. Derivative of exponential function

Differentiation rules

1. Derivative of the sum or difference
2. Derivative of a product
2a. Derivative of an expression multiplied by a constant factor
3. Derivative of the quotient
4. Derivative of a complex function

Rule 1If functions

are differentiable at some point , then at the same point the functions

and

those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.

Consequence. If two differentiable functions differ by a constant, then their derivatives are, i.e.

Rule 2If functions

are differentiable at some point , then their product is also differentiable at the same point

and

those. the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other.

Consequence 1. The constant factor can be taken out of the sign of the derivative:

Consequence 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors and all the others.

For example, for three multipliers:

Rule 3If functions

differentiable at some point and , then at this point their quotient is also differentiable.u/v , and

those. the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator.

Where to look on other pages

When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so more examples on these derivatives are in the article."The derivative of a product and a quotient".

Comment. You should not confuse a constant (that is, a number) as a term in the sum and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. This is typical mistake, which occurs on initial stage learning derivatives, but as they solve several one-two-component examples, the average student no longer makes this mistake.

And if, when differentiating a product or a quotient, you have a term u"v, wherein u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (such a case is analyzed in example 10).

Other common mistake- mechanical solution of the derivative of a complex function as a derivative of a simple function. So derivative of a complex function devoted to a separate article. But first we will learn to find derivatives simple functions.

Along the way, you can not do without transformations of expressions. To do this, you may need to open in new windows manuals Actions with powers and roots and Actions with fractions .

If you are looking for solutions to derivatives with powers and roots, that is, when the function looks like , then follow the lesson " Derivative of the sum of fractions with powers and roots".

If you have a task like , then you are in the lesson "Derivatives of simple trigonometric functions".

Step by step examples - how to find the derivative

Example 3 Find the derivative of a function

Decision. We determine the parts of the expression of the function: the entire expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the product differentiation rule: the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other:

Next, we apply the rule of differentiation of the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum, we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values ​​of derivatives:

We substitute the found derivatives into the sum of products and obtain the derivative of the entire function required by the condition of the problem:

Example 4 Find the derivative of a function

Decision. We are required to find the derivative of the quotient. We apply the formula for differentiating a quotient: the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator. We get:

We have already found the derivative of the factors in the numerator in Example 2. Let's also not forget that the product, which is the second factor in the numerator, is taken with a minus sign in the current example:

If you are looking for solutions to such problems in which you need to find the derivative of a function, where there is a continuous pile of roots and degrees, such as, for example, then welcome to class "The derivative of the sum of fractions with powers and roots" .

If you need to learn more about the derivatives of sines, cosines, tangents and other trigonometric functions, that is, when the function looks like , then you have a lesson "Derivatives of simple trigonometric functions" .

Example 5 Find the derivative of a function

Decision. In this function, we see a product, one of the factors of which is the square root of the independent variable, with the derivative of which we familiarized ourselves in the table of derivatives. According to the product differentiation rule and the tabular value of the derivative of the square root, we get:

Example 6 Find the derivative of a function

Decision. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the tabular value of the derivative of the square root, we get:

To get rid of the fraction in the numerator, multiply the numerator and denominator by .