The problem of finding the derivative of a given function is one of the main ones in the course of mathematics in high school and in higher educational institutions. It is impossible to fully explore a function, build its graph without taking its derivative. The derivative of a function can be easily found if you know the basic rules of differentiation, as well as the table of derivatives of the main functions. Let's figure out how to find the derivative of a function.

The derivative of a function is called the limit of the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero.

It is rather difficult to understand this definition, since the concept of a limit is not fully studied at school. But in order to find derivatives of various functions, it is not necessary to understand the definition, let's leave it to mathematicians and go straight to finding the derivative.

The process of finding the derivative is called differentiation. When differentiating a function, we will get a new function.

For their designation, we will use the Latin letters f, g, etc.

There are many different notations for derivatives. We will use stroke. For example, the entry g" means that we will find the derivative of the function g.

Derivative table

In order to answer the question of how to find the derivative, it is necessary to provide a table of derivatives of the main functions. To calculate the derivatives of elementary functions, it is not necessary to perform complex calculations. It is enough just to look at its value in the table of derivatives.

1. (sinx)"=cosx
2. (cos x)"= -sin x
3. (xn)"=nxn-1
4. (ex)"=ex
5. (lnx)"=1/x
6. (a x)"=a x ln a
7. (log a x)"=1/x ln a
8. (tg x)"=1/cos 2 x
9. (ctg x)"= - 1/sin 2 x
10. (arcsin x)"= 1/√(1-x 2)
11. (arccos x)"= - 1/√(1-x 2)
12. (arctg x)"= 1/(1+x 2)
13. (arcctg x)"= - 1/(1+x 2)

Example 1. Find the derivative of the function y=500.

We see that it is a constant. According to the table of derivatives, it is known that the derivative of the constant is equal to zero (formula 1).

Example 2. Find the derivative of the function y=x 100 .

This is a power function whose exponent is 100, and to find its derivative, you need to multiply the function by the exponent and lower it by 1 (formula 3).

(x 100)"=100 x 99

Example 3. Find the derivative of the function y=5 x

This is an exponential function, we calculate its derivative using formula 4.

Example 4. Find the derivative of the function y= log 4 x

We find the derivative of the logarithm using formula 7.

(log 4 x)"=1/x log 4

Differentiation rules

Let's now figure out how to find the derivative of a function if it is not in the table. Most of the investigated functions are not elementary, but are combinations of elementary functions using the simplest operations (addition, subtraction, multiplication, division, and multiplication by a number). To find their derivatives, you need to know the rules of differentiation. Further, the letters f and g denote functions, and C is a constant.

1. A constant coefficient can be taken out of the sign of the derivative

Example 5. Find the derivative of the function y= 6*x 8

We take out the constant coefficient 6 and differentiate only x 4 . This is a power function, the derivative of which we find according to formula 3 of the table of derivatives.

(6*x 8)" = 6*(x 8)"=6*8*x 7 =48* x 7

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

(f + g)"=f" + g"

Example 6. Find the derivative of the function y= x 100 + sin x

The function is the sum of two functions whose derivatives we can find from the table. Since (x 100)"=100 x 99 and (sin x)"=cos x. The derivative of the sum will be equal to the sum of these derivatives:

(x 100 + sin x)"= 100 x 99 + cos x

3. The derivative of the difference is equal to the difference of the derivatives

(f – g)"=f" – g"

Example 7. Find the derivative of the function y= x 100 - cos x

This function is the difference of two functions whose derivatives we can also find from the table. Then the derivative of the difference is equal to the difference of the derivatives and do not forget to change the sign, since (cos x) "= - sin x.

(x 100 - cos x) "= 100 x 99 + sin x

Example 8. Find the derivative of the function y=e x +tg x– x 2 .

This function has both a sum and a difference, we find the derivatives of each term:

(e x)"=e x, (tg x)"=1/cos 2 x, (x 2)"=2 x. Then the derivative of the original function is:

(e x +tg x– x 2)"= e x +1/cos 2 x –2 x

4. Derivative of a product

(f * g)"=f" * g + f * g"

Example 9. Find the derivative of the function y= cos x *e x

To do this, first find the derivative of each factor (cos x)"=–sin x and (e x)"=e x . Now let's substitute everything into the product formula. Multiply the derivative of the first function by the second and add the product of the first function by the derivative of the second.

(cos x* e x)"= e x cos x – e x *sin x

5. Derivative of the quotient

(f / g) "= f" * g - f * g "/ g 2

Example 10. Find the derivative of the function y= x 50 / sin x

To find the derivative of the quotient, first find the derivative of the numerator and denominator separately: (x 50)"=50 x 49 and (sin x)"= cos x. Substituting in the formula for the derivative of the quotient we get:

(x 50 / sin x) "= 50x 49 * sin x - x 50 * cos x / sin 2 x

Derivative of a complex function

A complex function is a function represented by a composition of several functions. To find the derivative of a complex function, there is also a rule:

(u(v))"=u"(v)*v"

Let's see how to find the derivative of such a function. Let y= u(v(x)) be a complex function. The function u will be called external, and v - internal.

For example:

y=sin (x 3) is a complex function.

Then y=sin(t) is the outer function

t=x 3 - internal.

Let's try to calculate the derivative of this function. According to the formula, it is necessary to multiply the derivatives of the inner and outer functions.

(sin t)"=cos (t) - derivative of the outer function (where t=x 3)

(x 3)"=3x 2 - derivative of the inner function

Then (sin (x 3))"= cos (x 3)* 3x 2 is the derivative of the complex function.

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. It's 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) and so on.

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.

• Table of derivatives of exponential and logarithmic functions

Derivatives of simple functions

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 With 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 With.

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).

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)

The process of finding the derivative of a function is called differentiation. The derivative has to be found in a number of problems in the course of mathematical analysis. For example, when finding extremum points and inflection points of a function graph.

How to find?

To find the derivative of a function, you need to know the table of derivatives of elementary functions and apply the basic rules of differentiation:

1. Taking the constant out of the sign of the derivative: $$ (Cu)" = C(u)" $$
2. Derivative of sum/difference of functions: $$ (u \pm v)" = (u)" \pm (v)" $$
3. Derivative of the product of two functions: $$ (u \cdot v)" = u"v + uv" $$
4. Fraction derivative : $$ \bigg (\frac(u)(v) \bigg)" = \frac(u"v - uv")(v^2) $$
5. Compound function derivative : $$ (f(g(x)))" = f"(g(x)) \cdot g"(x) $$

Solution examples

Derivative

Calculating the derivative of a mathematical function (differentiation) is a very common task in solving higher mathematics. For simple (elementary) mathematical functions, this is a fairly simple matter, since tables of derivatives for elementary functions have long been compiled and are easily accessible. However, finding the derivative of a complex mathematical function is not a trivial task and often requires significant effort and time.

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