If you follow the definition, then the derivative of a function at a point is the limit of the ratio of the increment of the function Δ y to the argument increment Δ x:

Everything seems to be clear. But try using this formula to calculate, say, the derivative of the function f(x) = x 2 + (2x+ 3) · e x sin x. If you do everything by definition, then after a couple of pages of calculations you will simply fall asleep. Therefore, there are simpler and more effective ways.

To begin with, we note that from the entire variety of functions we can distinguish the so-called elementary functions. These are relatively simple expressions, the derivatives of which have long been calculated and tabulated. Such functions are quite easy to remember - along with their derivatives.

Derivatives of elementary functions

Elementary functions are all those listed below. The derivatives of these functions must be known by heart. Moreover, it is not at all difficult to memorize them - that’s why they are elementary.

So, derivatives of elementary functions:

If an elementary function is multiplied by an arbitrary constant, then the derivative of the new function is also easily calculated:

(C · f)’ = C · f ’.

In general, constants can be taken out of the sign of the derivative. For example:

(2x 3)’ = 2 · ( x 3)’ = 2 3 x 2 = 6x 2 .

Obviously, elementary functions can be added to each other, multiplied, divided - and much more. This is how new functions will appear, no longer particularly elementary, but also differentiated according to certain rules. These rules are discussed below.

Derivative of sum and difference

Let the functions be given f(x) And g(x), the derivatives of which are known to us. For example, you can take the elementary functions discussed above. Then you can find the derivative of the sum and difference of these functions:

1. (f + g)’ = f ’ + g ’
2. (f − g)’ = f ’ − g ’

So, the derivative of the sum (difference) of two functions is equal to the sum (difference) of the derivatives. There may be more terms. For example, ( f + g + h)’ = f ’ + g ’ + h ’.

Strictly speaking, there is no concept of “subtraction” in algebra. There is a concept of “negative element”. Therefore the difference f − g can be rewritten as a sum f+ (−1) g, and then only one formula remains - the derivative of the sum.

f(x) = x 2 + sin x; g(x) = x 4 + 2x 2 − 3.

Function f(x) is the sum of two elementary functions, therefore:

f ’(x) = (x 2 + sin x)’ = (x 2)’ + (sin x)’ = 2x+ cos x;

We reason similarly for the function g(x). Only there are already three terms (from the point of view of algebra):

g ’(x) = (x 4 + 2x 2 − 3)’ = (x 4 + 2x 2 + (−3))’ = (x 4)’ + (2x 2)’ + (−3)’ = 4x 3 + 4x + 0 = 4x · ( x 2 + 1).

Answer:
f ’(x) = 2x+ cos x;
g ’(x) = 4x · ( x 2 + 1).

Derivative of the product

Mathematics is a logical science, so many people believe that if the derivative of a sum is equal to the sum of derivatives, then the derivative of the product strike">equal to the product of derivatives. But screw you! The derivative of a product is calculated using a completely different formula. Namely:

(f · g) ’ = f ’ · g + f · g ’

The formula is simple, but it is often forgotten. And not only schoolchildren, but also students. The result is incorrectly solved problems.

Task. Find derivatives of functions: f(x) = x 3 cos x; g(x) = (x 2 + 7x− 7) · e x .

Function f(x) is the product of two elementary functions, so everything is simple:

f ’(x) = (x 3 cos x)’ = (x 3)’ cos x + x 3 (cos x)’ = 3x 2 cos x + x 3 (− sin x) = x 2 (3cos x − x sin x)

Function g(x) the first multiplier is a little more complicated, but the general scheme does not change. Obviously, the first factor of the function g(x) is a polynomial and its derivative is the derivative of the sum. We have:

g ’(x) = ((x 2 + 7x− 7) · e x)’ = (x 2 + 7x− 7)’ · e x + (x 2 + 7x− 7) · ( e x)’ = (2x+ 7) · e x + (x 2 + 7x− 7) · e x = e x· (2 x + 7 + x 2 + 7x −7) = (x 2 + 9x) · e x = x(x+ 9) · e x .

Answer:
f ’(x) = x 2 (3cos x − x sin x);
g ’(x) = x(x+ 9) · e x .

Please note that in the last step the derivative is factorized. Formally, this does not need to be done, but most derivatives are not calculated on their own, but to examine the function. This means that further the derivative will be equated to zero, its signs will be determined, and so on. For such a case, it is better to have an expression factorized.

If there are two functions f(x) And g(x), and g(x) ≠ 0 on the set we are interested in, we can define a new function h(x) = f(x)/g(x). For such a function you can also find the derivative:

Not weak, huh? Where did the minus come from? Why g 2? And like this! This is one of the most complex formulas - you can’t figure it out without a bottle. Therefore, it is better to study it with specific examples.

Task. Find derivatives of functions:

The numerator and denominator of each fraction contain elementary functions, so all we need is the formula for the derivative of the quotient:

According to tradition, let's factorize the numerator - this will greatly simplify the answer:

A complex function is not necessarily a half-kilometer-long formula. For example, it is enough to take the function f(x) = sin x and replace the variable x, say, on x 2 + ln x. It will work out f(x) = sin ( x 2 + ln x) - this is a complex function. It also has a derivative, but it will not be possible to find it using the rules discussed above.

What should I do? In such cases, replacing a variable and formula for the derivative of a complex function helps:

f ’(x) = f ’(t) · t', If x is replaced by t(x).

As a rule, the situation with understanding this formula is even more sad than with the derivative of the quotient. Therefore, it is also better to explain it using specific examples, with a detailed description of each step.

Task. Find derivatives of functions: f(x) = e 2x + 3 ; g(x) = sin ( x 2 + ln x)

Note that if in the function f(x) instead of expression 2 x+ 3 will be easy x, then we get an elementary function f(x) = e x. Therefore, we make a replacement: let 2 x + 3 = t, f(x) = f(t) = e t. We look for the derivative of a complex function using the formula:

f ’(x) = f ’(t) · t ’ = (e t)’ · t ’ = e t · t ’

And now - attention! We perform the reverse replacement: t = 2x+ 3. We get:

f ’(x) = e t · t ’ = e 2x+ 3 (2 x + 3)’ = e 2x+ 3 2 = 2 e 2x + 3

Now let's look at the function g(x). Obviously it needs to be replaced x 2 + ln x = t. We have:

g ’(x) = g ’(t) · t’ = (sin t)’ · t’ = cos t · t ’

Reverse replacement: t = x 2 + ln x. Then:

g ’(x) = cos ( x 2 + ln x) · ( x 2 + ln x)’ = cos ( x 2 + ln x) · (2 x + 1/x).

That's all! As can be seen from the last expression, the whole problem has been reduced to calculating the derivative sum.

Answer:
f ’(x) = 2 · e 2x + 3 ;
g ’(x) = (2x + 1/x) cos ( x 2 + ln x).

Very often in my lessons, instead of the term “derivative,” I use the word “prime.” For example, the stroke of the sum is equal to the sum of the strokes. Is that clearer? Well, that's good.

Thus, calculating the derivative comes down to getting rid of these same strokes according to the rules discussed above. As a final example, let's return to the derivative power with a rational exponent:

(x n)’ = n · x n − 1

Few people know that in the role n may well be a fractional number. For example, the root is x 0.5. What if there is something fancy under the root? Again, the result will be a complex function - they like to give such constructions in tests and exams.

Task. Find the derivative of the function:

First, let's rewrite the root as a power with a rational exponent:

f(x) = (x 2 + 8x − 7) 0,5 .

Now we make a replacement: let x 2 + 8x − 7 = t. We find the derivative using the formula:

f ’(x) = f ’(t) · t ’ = (t 0.5)’ · t’ = 0.5 · t−0.5 · t ’.

Let's do the reverse replacement: t = x 2 + 8x− 7. We have:

f ’(x) = 0.5 · ( x 2 + 8x− 7) −0.5 · ( x 2 + 8x− 7)’ = 0.5 · (2 x+ 8) ( x 2 + 8x − 7) −0,5 .

Finally, back to the roots:

The calculator calculates the derivatives of all elementary functions, giving a detailed solution. The differentiation variable is determined automatically.

Derivative of a function- one of the most important concepts in mathematical analysis. The emergence of the derivative was led to such problems as, for example, calculating the instantaneous speed of a point at a moment in time, if the path depending on time is known, the problem of finding the tangent to a function at a point.

Most often, the derivative of a function is defined as the limit of the ratio of the increment of the function to the increment of the argument, if it exists.

Definition. Let the function be defined in some neighborhood of the point. Then the derivative of the function at a point is called the limit, if it exists

How to calculate the derivative of a function?

In order to learn to differentiate functions, you need to learn and understand differentiation rules and learn to use table of derivatives.

Rules of differentiation

Let and be arbitrary differentiable functions of a real variable and be some real constant. Then

— rule for differentiating the product of functions

— rule for differentiation of quotient functions

0 height=33 width=370 style="vertical-align: -12px;"> — differentiation of a function with a variable exponent

— rule for differentiating a complex function

— rule for differentiating a power function

Derivative of a function online

Our calculator will quickly and accurately calculate the derivative of any function online. The program will not make mistakes when calculating the derivative and will help you avoid long and tedious calculations. An online calculator will also be useful in cases where there is a need to check whether your solution is correct, and if it is incorrect, quickly find an error.

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 formulas for school mathematics course. The reference manual can be opened or downloaded on the page Mathematical formulas and tables. Also from there we will need Derivatives table, it is better to print it out; you will often have to refer to it, not only now, but also offline.

Eat? Let's get started. I have two news for you: good and very good. The good news is this: to learn how to find derivatives, you don’t have to know and understand what a derivative is. Moreover, it is more expedient to digest the definition of the derivative of a function, the mathematical, physical, geometric meaning of the derivative later, since a high-quality 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 same 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 recommend the following order of studying the topic:: First, this article. Then you need to read the most important lesson Derivative of a complex function. These two basic classes will take your skills from scratch. Next you can get acquainted with more complex derivatives in the article Complex derivatives. Logarithmic derivative. If the bar is too high, read the thing first The simplest typical problems with derivatives. In addition to the new material, the lesson covers other, simpler types of derivatives, and is a great opportunity to improve your differentiation technique. In addition, test papers almost always contain tasks on finding derivatives of functions that are specified implicitly or parametrically. There is also such a lesson: 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, let’s immediately look at an example:

Example 1

Find the derivative of a function

Solution:

This is a simple 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.

To put it quite simply, in order to find the derivative of a function, 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 exception is the exponential function, which turns into itself. The operation of finding the derivative is called differentiation .

Designations: The derivative is denoted by or .

ATTENTION, IMPORTANT! Forgetting to put a stroke (where it is necessary), or to draw an extra stroke (where it is not necessary) - BIG 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 the constant:
, where is a constant number;

derivative of a power function:
, in particular: , , .

Why remember? This knowledge is basic knowledge about derivatives. And if you cannot answer the teacher’s question “What is the derivative of a number?”, then your studies at the university may end for you (I am personally familiar with two real life cases). In addition, these are the most common formulas that we have to use almost every time we come across derivatives.

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

In this regard, we move on to consider differentiation rules:

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

Where is a constant number (constant)

Example 2

Find the derivative of a function

Let's 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 the constant factor out of the sign of the derivative:

Now we convert our cosine according to the table:

Well, it’s advisable to “comb” the result a little - put the minus sign in 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

Example 3

Find the derivative of a function

Let's decide. As you probably already noticed, the first step that is always performed when finding a derivative is that we enclose the entire expression in parentheses and put a prime at the top right:

Let's apply the second rule:

Please note that for differentiation, all roots and degrees must be represented in the form, and if they are in the denominator, then move them up. How to do this is discussed in my teaching materials.

Now let’s remember the first rule of differentiation - we take the constant factors (numbers) outside the derivative sign:

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

All functions located under the strokes are elementary table functions; using the table we carry out the transformation:

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

It is advisable to represent all powers of the type again in the form of roots; powers with negative exponents should be reset to the denominator. Although you don't have to do this, it won't be a mistake.

Example 4

Find the derivative of a function

Try to solve this example yourself (answer at the end of the lesson). Those interested can also use intensive course in pdf format, which is especially relevant if you have very little time at your disposal.

3) Derivative of the product of functions

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

This is an unusual rule (as, in fact, others) follows from derivative definitions. But we’ll hold off on the theory for now – now it’s more important to learn how to solve:

Example 5

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 using the derivative table:

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

Example 6

Find the derivative of a function

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

It's the same here. AT 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 are left with only elementary functions; using the table of derivatives, we transform them into other functions:

Ready.

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

Example 7

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson)

4) Derivative of quotient functions

A hatch opened in the ceiling, don't be alarmed, it's a glitch.
But this is the harsh reality:

Example 8

Find the derivative of a function

What’s missing here – sum, difference, product, fraction…. What should I start with?! There are doubts, there are no doubts, but, ANYWAY First, draw brackets and put a stroke at the top right:

Now we look at the expression in brackets, how can we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to place outside the sign of the derivative.

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Let and be functions of the independent variable x. Let them be differentiable in some range of values ​​of the variable x. Then, in this area, the derivative of the sum (difference) of these functions is equal to the sum (difference) of the derivatives of these functions:
(1) .

Proof

Since the functions and are differentiable at , there are the following limits, which are derivatives of these functions:
;
.

Consider the function y of the variable x, which is the sum of the functions and:
.
Let's apply the definition of derivative.


.

Thus, we have proven that the derivative of the sum of functions is equal to the sum of the derivatives:
.

In the same way, you can show that the derivative of the difference of functions is equal to the difference of derivatives:
.
This can be shown in another way, using the just proven rule for differentiating the sum and :
.

These two rules can be written as one equation:
(1) .

Consequence

Above we looked at the rule for finding the derivative of the sum of two functions. This rule can be generalized to the sum and difference of any number of differentiable functions.

The derivative of the sum (difference) of any finite number of differentiable functions is equal to the sum (difference) of their derivatives. Taking into account the rule of placing a constant outside the sign of the derivative, this rule can be written as follows:
.
Or in expanded form:
(2) .
Here - constants;
- differentiable functions of the variable x.

Evidence of the investigation

When n = 2 , we apply rule (1) and the rule of placing the constant outside the sign of the derivative. We have:
.
When n = 3 apply formula (1) for functions and :
.

For an arbitrary number n, we apply the induction method. Let equation (2) be satisfied for . Then for we have:

.
That is, from the assumption that equation (2) holds for , it follows that equation (2) holds for . And since equation (2) is true for , it is true for all .
The investigation has been proven.

Examples

Example 1

Find the derivative
.

Solution

Opening the parentheses. To do this we apply the formula
.
We also use the properties of power functions.
;

;
.

We apply formula (2) for the derivative of the sum and difference of functions.
.

From the table of derivatives we find:
.
Then
;
;
.

Finally we have:
.

Answer

Example 2

Find the derivative of a function with respect to the variable x
.

Solution

Let us reduce the roots to power functions.
.
We apply the rule of differentiation of sum and difference.
.
We apply the formulas from the table of derivatives.
;
;
;
;
;
.
Let's substitute:
.
We bring fractions to a common denominator.
.
Here we took into account that the given function is defined at .
.

Answer

Example 3

Find the derivative of a function
.

Solution

Let's transform the function. To do this, we apply the properties of the power function and roots:

;
;
;
.

We find the derivative using rule (2):


.