When deriving the very first formula of the table, we will proceed from the definition of the derivative function at a point. Let's take where x– any real number, that is, x– any number from the domain of definition of the function. Let us write down the limit of the ratio of the increment of the function to the increment of the argument at :

It should be noted that under the limit sign the expression is obtained, which is not the uncertainty of zero divided by zero, since the numerator does not contain an infinitesimal value, but precisely zero. In other words, the increment of a constant function is always zero.

Thus, derivative of a constant functionis equal to zero throughout the entire domain of definition.

Derivative of a power function.

The formula for the derivative of a power function has the form , where the exponent p– any real number.

Let us first prove the formula for the natural exponent, that is, for p = 1, 2, 3, …

We will use the definition of derivative. Let us write down the limit of the ratio of the increment of a power function to the increment of the argument:

To simplify the expression in the numerator, we turn to the Newton binomial formula:

Hence,

This proves the formula for the derivative of a power function for a natural exponent.

Derivative of an exponential function.

We present the derivation of the derivative formula based on the definition:

We have arrived at uncertainty. To expand it, we introduce a new variable, and at . Then . In the last transition, we used the formula for transitioning to a new logarithmic base.

Let's substitute into the original limit:

If we recall the second remarkable limit, we arrive at the formula for the derivative of the exponential function:

Derivative of a logarithmic function.

Let us prove the formula for the derivative of a logarithmic function for all x from the domain of definition and all valid values ​​of the base a logarithm By definition of derivative we have:

As you noticed, during the proof the transformations were carried out using the properties of the logarithm. Equality is true due to the second remarkable limit.

Derivatives of trigonometric functions.

To derive formulas for derivatives of trigonometric functions, we will have to recall some trigonometry formulas, as well as the first remarkable limit.

By definition of the derivative for the sine function we have .

Let's use the difference of sines formula:

It remains to turn to the first remarkable limit:

Thus, the derivative of the function sin x There is cos x.

The formula for the derivative of the cosine is proved in exactly the same way.

Therefore, the derivative of the function cos x There is –sin x.

We will derive formulas for the table of derivatives for tangent and cotangent using proven rules of differentiation (derivative of a fraction).

Derivatives of hyperbolic functions.

The rules of differentiation and the formula for the derivative of the exponential function from the table of derivatives allow us to derive formulas for the derivatives of the hyperbolic sine, cosine, tangent and cotangent.

Derivative of the inverse function.

To avoid confusion during presentation, let's denote in subscript the argument of the function by which differentiation is performed, that is, it is the derivative of the function f(x) By x.

Now let's formulate rule for finding the derivative of an inverse function.

Let the functions y = f(x) And x = g(y) mutually inverse, defined on the intervals and respectively. If at a point there is a finite non-zero derivative of the function f(x), then at the point there is a finite derivative of the inverse function g(y), and . In another post .

This rule can be reformulated for any x from the interval , then we get .

Let's check the validity of these formulas.

Let's find the inverse function for the natural logarithm (Here y is a function, and x- argument). Having resolved this equation for x, we get (here x is a function, and y– her argument). That is, and mutually inverse functions.

From the table of derivatives we see that And .

Let’s make sure that the formulas for finding the derivatives of the inverse function lead us to the same results:

The operation of finding the 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. The first to work in the field of finding derivatives were Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716).

Therefore, in our time, to find the derivative of any function, you do not need to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but you 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 prime sign break down simple functions into components and determine what actions (product, sum, quotient) these functions are related. Next, 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 derivative table and differentiation rules are given after the first two examples.

Example 1. Find the derivative of a function

Solution. From the rules of differentiation we find out that the derivative of a 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 sine is equal to cosine. We substitute these values ​​into the sum of derivatives and find the derivative required by the condition of the problem:

Example 2. Find the derivative of a function

Solution. We differentiate as a derivative of a sum in which the second term has a constant factor; it can be taken out of the sign of the derivative:

If questions still arise about where something comes from, they are usually cleared up after familiarizing yourself with the table of derivatives and the simplest rules of differentiation. We are moving on 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 equal to 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 for a long time
3. Derivative of degree. When solving problems, you need to convert non-square roots into powers.
4. Derivative of a variable to the power -1
5. Derivative square root
6. Derivative of sine
7. Derivative of cosine
8. Derivative of tangent
9. Derivative of cotangent
10. Derivative of arcsine
11. Derivative of arccosine
12. Derivative of arctangent
13. Derivative of arc cotangent
14. Derivative of the natural logarithm
15. Derivative of a logarithmic function
16. Derivative of the exponent
17. Derivative of an exponential function

Rules of differentiation

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

Rule 1.If the functions

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

and

those. the derivative of an 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 term, then their derivatives are equal, i.e.

Rule 2.If the functions

are differentiable at some point, then their product is 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.

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

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

For example, for three multipliers:

Rule 3.If the functions

differentiable at some point And , then at this point their quotient is also differentiableu/v , and

those. the derivative of the quotient of two functions is equal to a fraction, the numerator of which 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 for things on other pages

When finding the derivative of a product and the quotient in real problems It is always necessary to apply several differentiation rules at once, therefore more examples for these derivatives - in the article"Derivative of the product and quotient of functions".

Comment. You should not confuse a constant (that is, a number) as a term in a 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 typical mistake, which occurs at the initial stage of studying derivatives, but as the average student solves several one- and two-part examples, he no longer makes this mistake.

And if, when differentiating a product or quotient, you have a term u"v, in which 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 (this case is discussed in example 10).

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

Along the way, you can’t do without transforming expressions. To do this, you may need to open the manual in new windows. Actions with powers and roots And Operations with fractions .

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

If you have a task like , then you will take the lesson “Derivatives of simple trigonometric functions”.

Step-by-step examples - how to find the derivative

Example 3. Find the derivative of a function

Solution. We define the parts of the function expression: the entire expression represents a 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 by 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 has 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 turns 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 obtain the following derivative values:

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

Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating the quotient: the derivative of the quotient of two functions is equal to a fraction, the numerator of which 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 problems in which you need to find the derivative of a function, where there is a continuous pile of roots and powers, such as, for example, , then welcome to class "Derivative of sums 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 a lesson for you "Derivatives of simple trigonometric functions" .

Example 5. Find the derivative of a function

Solution. In this function we see a product, one of the factors of which is the square root of the independent variable, the derivative of which we familiarized ourselves with in the table of derivatives. Using the rule for differentiating the product and the tabular value of the derivative of the square root, we obtain:

Example 6. Find the derivative of a function

Solution. In this function we see a quotient whose dividend is the square root of the independent variable. Using the rule of differentiation of quotients, which we repeated and applied in example 4, and the tabulated value of the derivative of the square root, we obtain:

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

Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in mathematical analysis. We decided to devote today’s article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of derivative

Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values ​​of a function at two points. Definition of derivative:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise it can be written like this:

What's the point of finding such a limit? And here's what it is:

the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.

Physical meaning of the derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.

Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . average speed for a certain period of time:

To find out the speed of movement at a moment in time t0 you need to calculate the limit:

Rule one: set a constant

The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of the function:

Rule three: derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.

In the above example we come across the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.

Rule four: derivative of the quotient of two functions

Formula for determining the derivative of the quotient of two functions:

We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

With any questions on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult test and understand the tasks, even if you have never done derivative calculations before.

What is a derivative?

Table of derivatives.

Derivative is one of the main concepts of higher mathematics. In this lesson we will introduce this concept. Let's get to know each other, without strict mathematical formulations and proofs.

This acquaintance will allow you to:

Understand the essence of simple tasks with derivatives;

Successfully solve these simplest tasks;

Prepare for more serious lessons on derivatives.

First - a pleasant surprise.)

The strict definition of the derivative is based on the theory of limits and the thing is quite complicated. This is upsetting. But the practical application of derivatives, 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. That's all. This makes me happy.

Let's start getting acquainted?)

Terms and designations.

There are many different mathematical operations in elementary mathematics. Addition, subtraction, multiplication, exponentiation, logarithm, etc. If you add one more operation 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.

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

Differentiation- action on a function.

Derivative- the result of this action.

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

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

The derivative is indicated by a dash at the top right of the function. Like this: y" or f"(x) or S"(t) and so on.

Reading igrek stroke, ef stroke from x, es stroke from te, well, you understand...)

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

Let's assume that we have learned to understand the tasks. All that’s left is to learn how to solve them.) Let me remind you once again: finding the derivative is transformation of a function according to certain rules. Surprisingly, there are very few of these rules.

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

1. Table of derivatives (differentiation formulas).

3. Derivative of a complex function.

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

Table of derivatives.

There are an infinite number of functions in the world. Among this variety, there are functions that are most important for practical application. These functions are found in all laws of nature. From these functions, like 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 derivative and the theory of limits, this is a rather labor-intensive thing. And mathematicians are people too, yes, yes!) So they simplified their (and us) life. They calculated the 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. On the left is an elementary function, on the right is its derivative.

	Function y	Derivative of function y y"
1	C (constant value)	C" = 0
2	x	x" = 1
3	x n (n - any number)	(x n)" = nx n-1
	x 2 (n = 2)	(x 2)" = 2x
4	sin x	(sin x)" = cosx
	cos x	(cos x)" = - sin x
	tg x
	ctg x
5	arcsin x
	arccos x
	arctan 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! Do you get the hint?) Yes, it is advisable 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 table 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 wording of the task, or in the original function, which doesn’t 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 derivative of the power function in general view(third group). In our case n=3. So we substitute three instead of n and carefully write down the result:

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

That's 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 into this same derivative. Exactly 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 a new function.

Using the tablet we find the sine and the corresponding derivative:

y" = (sin x)" = cosx

We substitute zero into the derivative:

y"(0) = cos 0 = 1

This will be the answer.

3. Differentiate the function:

What, does it inspire?) There is no such 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, looking for the derivative of our function is quite troublesome. The table doesn't help...

But if we see that our function is double angle cosine, then everything gets better right away!

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

Those. our tricky function is nothing more than y = cosx. And this is a table function. We immediately get:

Answer: y" = - sin x.

Example for advanced graduates and students:

4. Find the derivative of the function:

There is no such function in the derivatives table, of course. But if you remember elementary mathematics, operations 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! Third group, n=1/10. We write directly according to the formula:

That's all. This will be the answer.

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