How to find the derivative of a function in degree. Derivative of a logarithmic function

Proof and derivation of formulas for the derivative of the exponential (e to the power of x) and the exponential function (a to the power of x). Examples of calculating derivatives of e^2x, e^3x and e^nx. Formulas for derivatives of higher orders.

The derivative of the exponent is equal to the exponent itself (the derivative of e to the power of x is equal to e to the power of x):
(1) (e x )′ = e x.

The derivative of an exponential function with a base of degree a is equal to the function itself, multiplied by the natural logarithm of a:
(2) .

Derivation of the formula for the derivative of the exponent, e to the power of x

The exponent is an exponential function whose exponent base is equal to the number e, which is the following limit:
.
Here it can be either a natural or a real number. Next, we derive formula (1) for the derivative of the exponent.

Derivation of the formula for the derivative of the exponent

Consider the exponent, e to the power of x :
y = e x .
This function is defined for all . Let's find its derivative with respect to x . By definition, the derivative is the following limit:
(3) .

Let's transform this expression to reduce it to known mathematical properties and rules. For this we need the following facts:
BUT) Exponent property:
(4) ;
B) Logarithm property:
(5) ;
AT) Continuity of the logarithm and property of limits for a continuous function:
(6) .
Here, is some function that has a limit and this limit is positive.
G) The meaning of the second wonderful limit:
(7) .

We apply these facts to our limit (3). We use property (4):
;
.

Let's make a substitution. Then ; .
Due to the continuity of the exponent,
.
Therefore, at , . As a result, we get:
.

Let's make a substitution. Then . At , . And we have:
.

We apply the property of the logarithm (5):
. Then
.

Let us apply property (6). Since there is a positive limit and the logarithm is continuous, then:
.
Here we also used the second remarkable limit (7). Then
.

Thus, we have obtained formula (1) for the derivative of the exponent.

Derivation of the formula for the derivative of the exponential function

Now we derive the formula (2) for the derivative of the exponential function with a base of degree a. We believe that and . Then the exponential function
(8)
Defined for everyone.

Let us transform formula (8). For this we use properties of the exponential function and logarithm.
;
.
So, we have transformed formula (8) to the following form:
.

Higher order derivatives of e to the power of x

Now let's find derivatives of higher orders. Let's look at the exponent first:
(14) .
(1) .

We see that the derivative of the function (14) is equal to the function (14) itself. Differentiating (1), we obtain second and third order derivatives:
;
.

This shows that the nth order derivative is also equal to the original function:
.

Higher order derivatives of the exponential function

Now consider an exponential function with a base of degree a:
.
We found its first order derivative:
(15) .

Differentiating (15), we obtain second and third order derivatives:
;
.

We see that each differentiation leads to the multiplication of the original function by . Therefore, the nth derivative has the following form:
.

Definition of exponential function. Derivation of a formula for calculating its derivative. Examples of calculating derivatives of exponential functions are analyzed in detail.

exponential function is a function that has the form of a power function
y = u v ,
whose base u and exponent v are some functions of the variable x :
u = u (x); v=v (x).
This function is also called exponential-power or .

Note that the exponential function can be represented in exponential form:
.
Therefore, it is also called complex exponential function.

Calculation using the logarithmic derivative

Find the derivative of the exponential function
(2) ,
where and are functions of the variable .
To do this, we take the logarithm of equation (2), using the property of the logarithm:
.
Differentiate with respect to x :
(3) .
Apply rules for differentiating a complex function and works:
;
.

Substitute in (3):
.
From here
.

So, we found the derivative of the exponential function:
(1) .
If the exponent is constant, then . Then the derivative is equal to the derivative of the compound power function:
.
If the base of the degree is constant, then . Then the derivative is equal to the derivative of the compound exponential function:
.
When and are functions of x, then the derivative of the exponential function is equal to the sum of the derivatives of the compound power and exponential functions.

Calculation of the derivative by reduction to a complex exponential function

Now we find the derivative of the exponential function
(2) ,
representing it as a complex exponential function:
(4) .

Let's differentiate the product:
.
We apply the rule for finding the derivative of a complex function:

.
And we again got the formula (1).

Example 1

Find the derivative of the following function:
.

Decision

We calculate using the logarithmic derivative. We take the logarithm of the original function:
(P1.1) .

From the table of derivatives we find:
;
.
According to the formula for the derivative of a product, we have:
.
We differentiate (A1.1):
.
Insofar as
,
then
.

Answer

Example 2

Find the derivative of a function
.

Decision

We take the logarithm of the original function:
(P2.1) .

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 of the 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 a typical mistake that occurs at the initial stage of studying derivatives, but as the average student solves several one-two-component examples, this mistake no longer makes.

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).

Another common mistake is the mechanical solution of the derivative of a complex function as the derivative of a simple function. So derivative of a complex function devoted to a separate article. But first we will learn to find derivatives of 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 function expression: 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 in the current example, is taken with a minus sign:

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 .

On which we analyzed the simplest derivatives, and also got acquainted with the rules of differentiation and some techniques for finding derivatives. Thus, if you are not very good with derivatives of functions or some points of this article are not entirely clear, then first read the above lesson. Please tune in to a serious mood - the material is not easy, but I will still try to present it simply and clearly.

In practice, you have to deal with the derivative of a complex function very often, I would even say almost always, when you are given tasks to find derivatives.

We look in the table at the rule (No. 5) for differentiating a complex function:

We understand. First of all, let's take a look at the notation. Here we have two functions - and , and the function, figuratively speaking, is nested in the function . A function of this kind (when one function is nested within another) is called a complex function.

I will call the function external function, and the function – inner (or nested) function.

! These definitions are not theoretical and should not appear in the final design of assignments. I use the informal expressions "external function", "internal" function only to make it easier for you to understand the material.

To clarify the situation, consider:

Example 1

Find the derivative of a function

Under the sine, we have not just the letter "x", but the whole expression, so finding the derivative immediately from the table will not work. We also notice that it is impossible to apply the first four rules here, there seems to be a difference, but the fact is that it is impossible to “tear apart” the sine:

In this example, already from my explanations, it is intuitively clear that the function is a complex function, and the polynomial is an internal function (embedding), and an external function.

First step, which must be performed when finding the derivative of a complex function is to understand which function is internal and which is external.

In the case of simple examples, it seems clear that a polynomial is nested under the sine. But what if it's not obvious? How to determine exactly which function is external and which is internal? To do this, I propose to use the following technique, which can be carried out mentally or on a draft.

Let's imagine that we need to calculate the value of the expression with a calculator (instead of one, there can be any number).

What do we calculate first? Primarily you will need to perform the following action: , so the polynomial will be an internal function:

Secondly you will need to find, so the sine - will be an external function:

After we UNDERSTAND with inner and outer functions, it's time to apply the compound function differentiation rule .

We start to decide. From the lesson How to find the derivative? we remember that the design of the solution of any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:

At first we find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that . All tabular formulas are applicable even if "x" is replaced by a complex expression, in this case:

Note that the inner function has not changed, we do not touch it.

Well, it is quite obvious that

The result of applying the formula clean looks like this:

The constant factor is usually placed at the beginning of the expression:

If there is any misunderstanding, write down the decision on paper and read the explanations again.

Example 2

Find the derivative of a function

Example 3

Find the derivative of a function

As always, we write:

We figure out where we have an external function, and where is an internal one. To do this, we try (mentally or on a draft) to calculate the value of the expression for . What needs to be done first? First of all, you need to calculate what the base is equal to:, which means that the polynomial is the internal function:

And, only then exponentiation is performed, therefore, the power function is an external function:

According to the formula , first you need to find the derivative of the external function, in this case, the degree. We are looking for the desired formula in the table:. We repeat again: any tabular formula is valid not only for "x", but also for a complex expression. Thus, the result of applying the rule of differentiation of a complex function next:

I emphasize again that when we take the derivative of the outer function, the inner function does not change:

Now it remains to find a very simple derivative of the inner function and “comb” the result a little:

Example 4

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

To consolidate the understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, reason, where is the external and where is the internal function, why are the tasks solved that way?

Example 5

a) Find the derivative of a function

b) Find the derivative of the function

Example 6

Find the derivative of a function

Here we have a root, and in order to differentiate the root, it must be represented as a degree. Thus, we first bring the function into the proper form for differentiation:

Analyzing the function, we come to the conclusion that the sum of three terms is an internal function, and exponentiation is an external function. We apply the rule of differentiation of a complex function :

The degree is again represented as a radical (root), and for the derivative of the internal function, we apply a simple rule for differentiating the sum:

Ready. You can also bring the expression to a common denominator in brackets and write everything as one fraction. It’s beautiful, of course, but when cumbersome long derivatives are obtained, it’s better not to do this (it’s easy to get confused, make an unnecessary mistake, and it will be inconvenient for the teacher to check).

Example 7

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

It is interesting to note that sometimes, instead of the rule for differentiating a complex function, one can use the rule for differentiating a quotient , but such a solution will look like a perversion unusual. Here is a typical example:

Example 8

Find the derivative of a function

Here you can use the rule of differentiation of the quotient , but it is much more profitable to find the derivative through the rule of differentiation of a complex function:

We prepare the function for differentiation - we take out the minus sign of the derivative, and raise the cosine to the numerator:

Cosine is an internal function, exponentiation is an external function.
Let's use our rule :

We find the derivative of the inner function, reset the cosine back down:

Ready. In the considered example, it is important not to get confused in the signs. By the way, try to solve it with the rule , the answers must match.

Example 9

Find the derivative of a function

This is an example for self-solving (answer at the end of the lesson).

So far, we have considered cases where we had only one nesting in a complex function. In practical tasks, you can often find derivatives, where, like nesting dolls, one inside the other, 3 or even 4-5 functions are nested at once.

Example 10

Find the derivative of a function

We understand the attachments of this function. We try to evaluate the expression using the experimental value . How would we count on a calculator?

First you need to find, which means that the arcsine is the deepest nesting:

This arcsine of unity should then be squared:

And finally, we raise the seven to the power:

That is, in this example we have three different functions and two nestings, while the innermost function is the arcsine, and the outermost function is the exponential function.

We start to decide

According to the rule first you need to take the derivative of the outer function. We look at the table of derivatives and find the derivative of the exponential function: The only difference is that instead of "x" we have a complex expression, which does not negate the validity of this formula. So, the result of applying the rule of differentiation of a complex function next.

Derivation of the formula for the derivative of a power function (x to the power of a). Derivatives of roots from x are considered. The formula for the derivative of a higher order power function. Examples of calculating derivatives.

The derivative of x to the power of a is a times x to the power of a minus one:
(1) .

The derivative of the nth root of x to the mth power is:
(2) .

Derivation of the formula for the derivative of a power function

Case x > 0

Consider a power function of variable x with exponent a :
(3) .
Here a is an arbitrary real number. Let's consider the case first.

To find the derivative of the function (3), we use the properties of the power function and transform it to the following form:
.

Now we find the derivative by applying:
;
.
Here .

Formula (1) is proved.

Derivation of the formula for the derivative of the root of the degree n of x to the degree m

Now consider a function that is the root of the following form:
(4) .

To find the derivative, we convert the root to a power function:
.
Comparing with formula (3), we see that
.
Then
.

By formula (1) we find the derivative:
(1) ;
;
(2) .

In practice, there is no need to memorize formula (2). It is much more convenient to first convert the roots to power functions, and then find their derivatives using formula (1) (see examples at the end of the page).

Case x = 0

If , then the exponential function is also defined for the value of the variable x = 0 . Let us find the derivative of function (3) for x = 0 . To do this, we use the definition of a derivative:
.

Substitute x = 0 :
.
In this case, by derivative we mean the right-hand limit for which .

So we found:
.
From this it can be seen that at , .
At , .
At , .
This result is also obtained by formula (1):
(1) .
Therefore, formula (1) is also valid for x = 0 .

case x< 0

Consider function (3) again:
(3) .
For some values ​​of the constant a , it is also defined for negative values ​​of the variable x . Namely, let a be a rational number. Then it can be represented as an irreducible fraction:
,
where m and n are integers with no common divisor.

If n is odd, then the exponential function is also defined for negative values ​​of the variable x. For example, for n = 3 and m = 1 we have the cube root of x :
.
It is also defined for negative values ​​of x .

Let us find the derivative of the power function (3) for and for rational values ​​of the constant a , for which it is defined. To do this, we represent x in the following form:
.
Then ,
.
We find the derivative by taking the constant out of the sign of the derivative and applying the rule of differentiation of a complex function:

.
Here . But
.
Since , then
.
Then
.
That is, formula (1) is also valid for:
(1) .

Derivatives of higher orders

Now we find the higher order derivatives of the power function
(3) .
We have already found the first order derivative:
.

Taking the constant a out of the sign of the derivative, we find the second-order derivative:
.
Similarly, we find derivatives of the third and fourth orders:
;

.

From here it is clear that derivative of an arbitrary nth order has the following form:
.

notice, that if a is a natural number, , then the nth derivative is constant:
.
Then all subsequent derivatives are equal to zero:
,
at .

Derivative Examples

Example

Find the derivative of the function:
.

Decision

Let's convert the roots to powers:
;
.
Then the original function takes the form:
.

We find derivatives of degrees:
;
.
The derivative of a constant is zero:
.