Derivative calculation is one of the most important operations in differential calculus. Below is a table for finding derivatives of simple functions. For more complex differentiation rules, see other lessons:

• Table of derivatives of exponential and logarithmic functions

Use the given formulas as reference values. They will help in solving differential equations and problems. In the picture, in the table of derivatives of simple functions, there is a "cheat sheet" of the main cases of finding the derivative in a form that is understandable for use, next to it are explanations for each case.

Derivatives of simple functions

1. The derivative of a number is zero
с´ = 0
Example:
5' = 0

Explanation:
The derivative shows the rate at which the value of the function changes when the argument changes. Since the number does not change in any way under any conditions, the rate of its change is always zero.

2. Derivative of a variable equal to one
x' = 1

Explanation:
With each increment of the argument (x) by one, the value of the function (calculation result) increases by the same amount. Thus, the rate of change of the value of the function y = x is exactly equal to the rate of change of the value of the argument.

3. The derivative of a variable and a factor is equal to this factor
сx´ = с
Example:
(3x)´ = 3
(2x)´ = 2
Explanation:
In this case, each time the function argument ( X) its value (y) grows in from once. Thus, the rate of change of the value of the function with respect to the rate of change of the argument is exactly equal to the value from.

Whence it follows that
(cx + b)" = c
that is, the differential of the linear function y=kx+b is equal to the slope of the straight line (k).

4. Modulo derivative of a variable is equal to the quotient of this variable to its modulus
|x|"= x / |x| provided that x ≠ 0
Explanation:
Since the derivative of the variable (see formula 2) is equal to one, the derivative of the module differs only in that the value of the rate of change of the function changes to the opposite when crossing the origin point (try to draw a graph of the function y = |x| and see for yourself. This is exactly value and returns the expression x / |x| When x< 0 оно равно (-1), а когда x >0 - one. That is, with negative values ​​of the variable x, with each increase in the change in the argument, the value of the function decreases by exactly the same value, and with positive values, on the contrary, it increases, but by exactly the same value.

5. Power derivative of a variable is equal to the product of the number of this power and the variable in the power, reduced by one
(x c)"= cx c-1, provided that x c and cx c-1 are defined and c ≠ 0
Example:
(x 2)" = 2x
(x 3)" = 3x 2
To memorize the formula:
Take the exponent of the variable "down" as a multiplier, and then decrease the exponent itself by one. For example, for x 2 - two was ahead of x, and then the reduced power (2-1 = 1) just gave us 2x. The same thing happened for x 3 - we lower the triple, reduce it by one and instead of a cube we have a square, that is, 3x 2 . A little "unscientific", but very easy to remember.

6.Fraction derivative 1/x
(1/x)" = - 1 / x 2
Example:
Since a fraction can be represented as raising to a negative power
(1/x)" = (x -1)" , then you can apply the formula from rule 5 of the derivatives table
(x -1)" = -1x -2 = - 1 / x 2

7. Fraction derivative with a variable of arbitrary degree in the denominator
(1/x c)" = - c / x c+1
Example:
(1 / x 2)" = - 2 / x 3

8. root derivative(derivative of variable under square root)
(√x)" = 1 / (2√x) or 1/2 x -1/2
Example:
(√x)" = (x 1/2)" so you can apply the formula from rule 5
(x 1/2)" \u003d 1/2 x -1/2 \u003d 1 / (2√x)

9. Derivative of a variable under a root of an arbitrary degree
(n √ x)" = 1 / (n n √ x n-1)

Date: 11/20/2014

What is a derivative?

Derivative table.

The derivative is one of the main concepts of higher mathematics. In this lesson, we will introduce this concept. Let's get acquainted, without strict mathematical formulations and proofs.

This introduction will allow you to:

Understand the essence of simple tasks with a derivative;

Successfully solve these very simple tasks;

Prepare for more serious derivative lessons.

First, a pleasant surprise.

The strict definition of the derivative is based on the theory of limits, and the thing is rather complicated. It's upsetting. But the practical application of the derivative, as a rule, does not require such extensive and deep knowledge!

To successfully complete most tasks at school and university, it is enough to know just a few terms- to understand the task, and just a few rules- to solve it. And that's it. This makes me happy.

Shall we get to know each other?)

Terms and designations.

There are many mathematical operations in elementary mathematics. Addition, subtraction, multiplication, exponentiation, logarithm, etc. If one more operation is added to these operations, elementary mathematics becomes higher. This new operation is called differentiation. The definition and meaning of this operation will be discussed in separate lessons.

Here it is important to understand that differentiation is just a mathematical operation on a function. We take any function and, according to certain rules, transform it. The result is a new function. This new function is called: derivative.

Differentiation- action on a function.

Derivative is the result of this action.

Just like, for example, sum is the result of the addition. Or private is the result of the division.

Knowing the terms, you can at least understand the tasks.) The wording is as follows: find the derivative of a function; take the derivative; differentiate the function; calculate derivative etc. This is all same. Of course, there are more complex tasks, where finding the derivative (differentiation) will be just one of the steps in solving the task.

The derivative is denoted by a dash at the top right above the function. Like this: y" or f"(x) or S"(t) etc.

read y stroke, ef stroke from x, es stroke from te, well you get it...)

A prime can also denote the derivative of a particular function, for example: (2x+3)", (x 3 )" , (sinx)" etc. Often the derivative is denoted using differentials, but we will not consider such a notation in this lesson.

Suppose that we have learned to understand the tasks. There is nothing left - to learn how to solve them.) Let me remind you again: finding the derivative is transformation of a function according to certain rules. These rules are surprisingly few.

To find the derivative of a function, you only need to know three things. Three pillars on which all differentiation rests. Here are the three whales:

1. Table of derivatives (differentiation formulas).

3. Derivative of a complex function.

Let's start in order. In this lesson, we will consider the table of derivatives.

Derivative table.

The world has an infinite number of functions. Among this set there are functions which are most important for practical application. These functions sit in all the laws of nature. From these functions, as from bricks, you can construct all the others. This class of functions is called elementary functions. It is these functions that are studied at school - linear, quadratic, hyperbola, etc.

Differentiation of functions "from scratch", i.e. based on the definition of the derivative and the theory of limits - a rather time-consuming thing. And mathematicians are people too, yes, yes!) So they simplified their lives (and us). They calculated derivatives of elementary functions before us. The result is a table of derivatives, where everything is ready.)

Here it is, this plate for the most popular functions. Left - elementary function, right - its derivative.

	Function y	Derivative of function y y"
1	C (constant)	C" = 0
2	x	x" = 1
3	x n (n is any number)	(x n)" = nx n-1
	x 2 (n = 2)	(x 2)" = 2x
4	sin x	(sinx)" = cosx
	cos x	(cos x)" = - sin x
	tg x
	ctg x
5	arcsin x
	arccos x
	arctg x
	arcctg x
4	a x
	e x
5	log a x
	ln x ( a = e)

I recommend paying attention to the third group of functions in this table of derivatives. The derivative of a power function is one of the most common formulas, if not the most common! Is the hint clear?) Yes, it is desirable to know the table of derivatives by heart. By the way, this is not as difficult as it might seem. Try to solve more examples, the table itself will be remembered!)

Finding the tabular value of the derivative, as you understand, is not the most difficult task. Therefore, very often in such tasks there are additional chips. Either in the formulation of the task, or in the original function, which does not seem to be in the table ...

Let's look at a few examples:

1. Find the derivative of the function y = x 3

There is no such function in the table. But there is a general derivative of the power function (third group). In our case, n=3. So we substitute the triple instead of n and carefully write down the result:

(x 3) " = 3 x 3-1 = 3x 2

That's all there is to it.

Answer: y" = 3x 2

2. Find the value of the derivative of the function y = sinx at the point x = 0.

This task means that you must first find the derivative of the sine, and then substitute the value x = 0 to this same derivative. It's in that order! Otherwise, it happens that they immediately substitute zero into the original function ... We are asked to find not the value of the original function, but the value its derivative. The derivative, let me remind you, is already a new function.

On the plate we find the sine and the corresponding derivative:

y" = (sinx)" = cosx

Substitute zero into the derivative:

y"(0) = cos 0 = 1

This will be the answer.

3. Differentiate the function:

What inspires?) There is not even close such a function in the table of derivatives.

Let me remind you that to differentiate a function is simply to find the derivative of this function. If you forget elementary trigonometry, finding the derivative of our function is quite troublesome. The table doesn't help...

But if we see that our function is cosine of a double angle, then everything immediately gets better!

Yes Yes! Remember that the transformation of the original function before differentiation quite acceptable! And it happens to make life a lot easier. According to the formula for the cosine of a double angle:

Those. our tricky function is nothing but y = cox. And this is a table function. We immediately get:

Answer: y" = - sin x.

Example for advanced graduates and students:

4. Find the derivative of a function:

There is no such function in the derivatives table, of course. But if you remember elementary mathematics, actions with powers... Then it is quite possible to simplify this function. Like this:

And x to the power of one tenth is already a tabular function! The third group, n=1/10. Directly according to the formula and write:

That's all. This will be the answer.

I hope that with the first whale of differentiation - the table of derivatives - everything is clear. It remains to deal with the two remaining whales. In the next lesson, we will learn the rules of differentiation.

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? First of all 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.

How to find the derivative, how to take the derivative? In this lesson, we will learn how to find derivatives of functions. But before studying this page, I strongly recommend that you familiarize yourself with the methodological material.Hot School Mathematics Formulas. The reference manual can be opened or downloaded from the page Mathematical formulas and tables . Also from there we needDerivative table, it is better to print it, you will often have to refer to it, and not only now, but also offline.

There is? Let's get started. I have two news for you: good and very good. The good news is that in order to learn how to find derivatives, it is not at all necessary to know and understand what a derivative is. Moreover, the definition of the derivative of a function, the mathematical, physical, geometric meaning of the derivative is more expedient to digest later, since the qualitative study of the theory, in my opinion, requires the study of a number of other topics, as well as some practical experience.

And now our task is to master these very derivatives technically. The very good news is that learning to take derivatives is not so difficult, there is a fairly clear algorithm for solving (and explaining) this task, integrals or limits, for example, are more difficult to master.

I advise the following order of study of the topic: first, This article. Then you need to read the most important lesson Derivative of a complex function . These two basic classes will allow you to raise your skills from scratch. Further, it will be possible to familiarize yourself with more complex derivatives in the article. complex derivatives.

logarithmic derivative. If the bar is too high, read the item first The simplest typical problems with a derivative. In addition to the new material, the lesson covered other, simpler types of derivatives, and there is a great opportunity to improve your differentiation technique. In addition, in control work, there are almost always tasks for finding derivatives of functions that are specified implicitly or parametrically. There is also a tutorial for this: Derivatives of implicit and parametrically defined functions.

I will try in an accessible form, step by step, to teach you how to find derivatives of functions. All information is presented in detail, in simple words.

Actually, immediately consider an example: Example 1

Find the derivative of a function Solution:

This is the simplest example, please find it in the table of derivatives of elementary functions. Now let's look at the solution and analyze what happened? And the following thing happened:

we had a function , which, as a result of the solution, turned into a function.

Quite simply, to find the derivative

functions, you need to turn it into another function according to certain rules . Look again at the table of derivatives - there functions turn into other functions. the only

the exception is the exponential function, which

turns into itself. The operation of finding the derivative is calleddifferentiation.

Notation: The derivative is denoted or.

ATTENTION, IMPORTANT! Forgetting to put a stroke (where necessary), or drawing an extra stroke (where it is not necessary) is a GREAT MISTAKE! A function and its derivative are two different functions!

Let's return to our table of derivatives. From this table it is desirable memorize: rules of differentiation and derivatives of some elementary functions, especially:

derivative of a constant:

Where is a constant number; derivative of a power function:

In particular:,,.

Why memorize? This knowledge is elementary knowledge about derivatives. And if you can’t answer the teacher’s question “What is the derivative of the number?”, Then your studies at the university may end for you (I personally know two real cases from life). In addition, these are the most common formulas that we have to use almost every time we encounter derivatives.

IN In reality, simple tabular examples are rare; usually, when finding derivatives, differentiation rules are used first, and then a table of derivatives of elementary functions.

IN In this regard, we turn to the considerationdifferentiation rules:

1) A constant number can (and should) be taken out of the sign of the derivative

Where is a constant number (constant) Example 2

Find the derivative of a function

We look at the table of derivatives. The derivative of the cosine is there, but we have .

It's time to use the rule, we take out the constant factor beyond the sign of the derivative:

And now we turn our cosine according to the table:

Well, it is desirable to “comb” the result a little - put the minus in the first place, at the same time getting rid of the brackets:

2) The derivative of the sum is equal to the sum of the derivatives

Find the derivative of a function

We decide. As you probably already noticed, the first action that is always performed when finding the derivative is that we put the whole expression in brackets and put a dash on the top right:

We apply the second rule:

Please note that for differentiation, all roots, degrees must be represented as , and if they are in the denominator, then

move them up. How to do this is discussed in my methodological materials.

Now we recall the first rule of differentiation - we take out the constant factors (numbers) outside the sign of the derivative:

Usually, during the solution, these two rules are applied simultaneously (so as not to rewrite a long expression once again).

All functions under the dashes are elementary table functions, using the table we perform the transformation:

You can leave everything in this form, since there are no more strokes, and the derivative has been found. However, expressions like this usually simplify:

It is desirable to represent all degrees of the species again as roots,

degrees with negative exponents - reset to the denominator. Although you can not do this, it will not be a mistake.

Find the derivative of a function

Try to solve this example yourself (answer at the end of the lesson).

3) Derivative of the product of functions

It seems that, by analogy, the formula suggests itself ...., but the surprise is that:

This unusual rule(as, in fact, others) follows from definitions of the derivative. But we will wait with the theory for now - now it is more important to learn how to solve:

Find the derivative of a function

Here we have the product of two functions depending on . First we apply our strange rule, and then we transform the functions according to the table of derivatives:

Difficult? Not at all, quite affordable even for a teapot.

Find the derivative of a function

This function contains the sum and product of two functions - the square trinomial and the logarithm. We remember from school that multiplication and division take precedence over addition and subtraction.

It's the same here. FIRST we use the product differentiation rule:

Now for the bracket we use the first two rules:

As a result of applying the rules of differentiation under the strokes, we have only elementary functions left, according to the table of derivatives we turn them into other functions:

With some experience in finding derivatives, simple derivatives do not seem to need to be described in such detail. In general, they are usually resolved verbally, and it is immediately recorded that .

Find the derivative of a function This is an example for self-solving (answer at the end of the lesson)

4) Derivative of private functions

A hatch has opened in the ceiling, don't be scared, it's a glitch. And here is the harsh reality:

Find the derivative of a function

What is not here - the sum, difference, product, fraction .... What should I start with?! There are doubts, no doubts, but, IN ANY CASE, first draw brackets and put a stroke at the top right:

Now we look at the expression in brackets, how would we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to take it out of the sign of the derivative:

At the same time, we get rid of the brackets in the numerator, which are no longer needed. Generally speaking, the constant factors in finding the derivative

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
.
Because , 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:
.

Solution

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:
.