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 studied 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 the 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 compound 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 compound function.

Definition. Let the function \(y = f(x) \) be defined in some interval containing the point \(x_0 \) inside. Let's increment \(\Delta x \) to the argument so as not to leave this interval. Find the corresponding increment of the function \(\Delta y \) (when passing from the point \(x_0 \) to the point \(x_0 + \Delta x \)) and compose the relation \(\frac(\Delta y)(\Delta x) \). If there is a limit of this relation at \(\Delta x \rightarrow 0 \), then the indicated limit is called derivative function\(y=f(x) \) at the point \(x_0 \) and denote \(f"(x_0) \).

$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x_0) $$

The symbol y is often used to denote the derivative. Note that y" = f(x) is a new function, but naturally associated with the function y = f(x), defined at all points x at which the above limit exists . This function is called like this: derivative of the function y \u003d f (x).

The geometric meaning of the derivative consists of the following. If a tangent that is not parallel to the y axis can be drawn to the graph of the function y \u003d f (x) at a point with the abscissa x \u003d a, then f (a) expresses the slope of the tangent:
\(k = f"(a)\)

Since \(k = tg(a) \), the equality \(f"(a) = tg(a) \) is true.

And now we interpret the definition of the derivative in terms of approximate equalities. Let the function \(y = f(x) \) have a derivative at a particular point \(x \):
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x) $$
This means that near the point x, the approximate equality \(\frac(\Delta y)(\Delta x) \approx f"(x) \), i.e. \(\Delta y \approx f"(x) \cdot\Deltax\). The meaningful meaning of the obtained approximate equality is as follows: the increment of the function is “almost proportional” to the increment of the argument, and the coefficient of proportionality is the value of the derivative at a given point x. For example, for the function \(y = x^2 \) the approximate equality \(\Delta y \approx 2x \cdot \Delta x \) is true. If we carefully analyze the definition of the derivative, we will find that it contains an algorithm for finding it.

Let's formulate it.

How to find the derivative of the function y \u003d f (x) ?

1. Fix value \(x \), find \(f(x) \)
2. Increment \(x \) argument \(\Delta x \), move to a new point \(x+ \Delta x \), find \(f(x+ \Delta x) \)
3. Find the function increment: \(\Delta y = f(x + \Delta x) - f(x) \)
4. Compose the relation \(\frac(\Delta y)(\Delta x) \)
5. Calculate $$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) $$
This limit is the derivative of the function at x.

If the function y = f(x) has a derivative at the point x, then it is called differentiable at the point x. The procedure for finding the derivative of the function y \u003d f (x) is called differentiation functions y = f(x).

Let us discuss the following question: how are the continuity and differentiability of a function at a point related?

Let the function y = f(x) be differentiable at the point x. Then a tangent can be drawn to the graph of the function at the point M (x; f (x)) and, recall, the slope of the tangent is equal to f "(x). Such a graph cannot "break" at the point M, i.e., the function must be continuous at x.

It was reasoning "on the fingers". Let us present a more rigorous argument. If the function y = f(x) is differentiable at the point x, then the approximate equality \(\Delta y \approx f"(x) \cdot \Delta x \) holds. zero, then \(\Delta y \) will also tend to zero, and this is the condition for the continuity of the function at a point.

So, if a function is differentiable at a point x, then it is also continuous at that point.

The converse is not true. For example: function y = |x| is continuous everywhere, in particular at the point x = 0, but the tangent to the graph of the function at the “joint point” (0; 0) does not exist. If at some point it is impossible to draw a tangent to the function graph, then there is no derivative at this point.

One more example. The function \(y=\sqrt(x) \) is continuous on the entire number line, including at the point x = 0. And the tangent to the graph of the function exists at any point, including at the point x = 0. But at this point the tangent coincides with the y-axis, i.e., it is perpendicular to the abscissa axis, its equation has the form x \u003d 0. There is no slope for such a straight line, which means that \ (f "(0) \) does not exist either

So, we got acquainted with a new property of a function - differentiability. How can you tell if a function is differentiable from the graph of a function?

The answer is actually given above. If at some point a tangent can be drawn to the graph of a function that is not perpendicular to the x-axis, then at this point the function is differentiable. If at some point the tangent to the graph of the function does not exist or it is perpendicular to the x-axis, then at this point the function is not differentiable.

Differentiation rules

The operation of finding the derivative is called differentiation. When performing this operation, you often have to work with quotients, sums, products of functions, as well as with "functions of functions", that is, complex functions. Based on the definition of the derivative, we can derive differentiation rules that facilitate this work. If C is a constant number and f=f(x), g=g(x) are some differentiable functions, then the following are true differentiation rules:

$$ C"=0 $$ $$ x"=1 $$ $$ (f+g)"=f"+g" $$ $$ (fg)"=f"g + fg" $$ $$ ( Cf)"=Cf" $$ $$ \left(\frac(f)(g) \right) " = \frac(f"g-fg")(g^2) $$ $$ \left(\frac (C)(g) \right) " = -\frac(Cg")(g^2) $$ Compound function derivative:
$$ f"_x(g(x)) = f"_g \cdot g"_x $$

Table of derivatives of some functions

$$ \left(\frac(1)(x) \right) " = -\frac(1)(x^2) $$ $$ (\sqrt(x)) " = \frac(1)(2\ sqrt(x)) $$ $$ \left(x^a \right) " = a x^(a-1) $$ $$ \left(a^x \right) " = a^x \cdot \ln a $$ $$ \left(e^x \right) " = e^x $$ $$ (\ln x)" = \frac(1)(x) $$ $$ (\log_a x)" = \frac (1)(x\ln a) $$ $$ (\sin x)" = \cos x $$ $$ (\cos x)" = -\sin x $$ $$ (\text(tg) x) " = \frac(1)(\cos^2 x) $$ $$ (\text(ctg) x)" = -\frac(1)(\sin^2 x) $$ $$ (\arcsin x) " = \frac(1)(\sqrt(1-x^2)) $$ $$ (\arccos x)" = \frac(-1)(\sqrt(1-x^2)) $$ $$ (\text(arctg) x)" = \frac(1)(1+x^2) $$ $$ (\text(arctg) x)" = \frac(-1)(1+x^2) $ $

It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The derivative is one of the most important concepts of 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 the derivative

Let there be a function f(x) , given in some interval (a,b) . The points x and x0 belong to this interval. When x changes, the function itself also changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values ​​of the function at two points. Derivative definition:

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 is the point in finding such a limit? But which one:

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.

The physical meaning of the derivative: the time derivative of the path is equal to the speed of the rectilinear motion.

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

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

Rule one: take out the constant

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

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 a function:

Rule three: the 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:

Here it is important to say about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.

In the above example, we encounter 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 consider 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: The derivative of the quotient of two functions

Formula for determining the derivative of a 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 question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.

Date: 05/10/2015

How to find the derivative?

Differentiation rules.

To find the derivative of any function, you need to master only three concepts:

2. Rules of differentiation.

3. Derivative of a complex function.

Exactly in that order. It's a hint.)

Of course, it would be nice to have an idea about the derivative in general). About what a derivative is and how to work with a table of derivatives - it is accessible in the previous lesson. Here we will deal with the rules of differentiation.

Differentiation is the operation of finding a derivative. There is nothing more behind this term. Those. expressions "find the derivative of a function" and "differentiate function"- This is the same.

Expression "rules of differentiation" refers to finding the derivative from arithmetic operations. This understanding helps a lot to avoid porridge in the head.

Let's concentrate and remember all-all-all arithmetic operations. There are four of them). Addition (sum), subtraction (difference), multiplication (product), and division (quotient). Here they are, the rules of differentiation:

The plate shows five rules on four arithmetic operations. I didn't miscalculate.) It's just that rule 4 is an elementary corollary of rule 3. But it's so popular that it makes sense to write it down (and remember!) as an independent formula.

Under the notation U and V some (absolutely any!) functions are implied U(x) and V(x).

Let's look at a few examples. First, the simplest ones.

Find the derivative of the function y=sinx - x 2

Here we have difference two elementary functions. We apply rule 2. We will assume that sinx is a function U, and x 2 is a function v. We have every right to write:

y" = (sinx - x 2)" = (sinx)"- (x 2)"

Already better, right?) It remains to find the derivatives of the sine and the square of x. There is a derivative table for this. We just look in the table for the functions we need ( sinx and x 2), look at their derivatives and write down the answer:

y" = (sinx)" - (x 2)" = cosx - 2x

That's all there is to it. Rule 1 of differentiating the sum works in exactly the same way.

What if we have multiple terms? It's okay.) We break the function into terms and look for the derivative of each term, regardless of the others. For example:

Find the derivative of the function y=sinx - x 2 +cosx - x +3

Feel free to write:

y" = (sinx)" - (x 2)" + (cosx)" - (x)" + (3)"

At the end of the lesson I will give tips on making life easier when differentiating.)

Practical Tips:

1. Before differentiation, we look to see if it is possible to simplify the original function.

2. In confused examples, we paint the solution in detail, with all brackets and strokes.

3. When differentiating fractions with a constant number in the denominator, we turn division into multiplication and use rule 4.

The operation of finding a derivative is called differentiation.

As a result of solving problems of finding derivatives of the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716) were the first to work in the field of finding derivatives.

Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but only need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.

To find the derivative, you need an expression under the stroke sign break down simple functions and determine what actions (product, sum, quotient) these functions are related. Further, we find the derivatives of elementary functions in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient - in the rules of differentiation. The table of derivatives and differentiation rules are given after the first two examples.

Example 1 Find the derivative of a function

Solution. From the rules of differentiation we find out that the derivative of the sum of functions is the sum of derivatives of functions, i.e.

From the table of derivatives, we find out that the derivative of "X" is equal to one, and the derivative of the sine is cosine. We substitute these values ​​in the sum of derivatives and find the derivative required by the condition of the problem:

Example 2 Find the derivative of a function

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

If there are still questions about where something comes from, they, as a rule, become clear after reading the table of derivatives and the simplest rules of differentiation. We are going to them right now.

Table of derivatives of simple functions

1. Derivative of a constant (number). Any number (1, 2, 5, 200...) that is in the function expression. Always zero. This is very important to remember, as it is required very often
2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember
3. Derivative of degree. When solving problems, you need to convert non-square roots to a power.
4. Derivative of a variable to the power of -1
5. Derivative of the square root
6. Sine derivative
7. Cosine derivative
8. Tangent derivative
9. Derivative of cotangent
10. Derivative of the arcsine
11. Derivative of arc cosine
12. Derivative of arc tangent
13. Derivative of the inverse tangent
14. Derivative of natural logarithm
15. Derivative of a logarithmic function
16. Derivative of the exponent
17. Derivative of exponential function

Differentiation rules

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

Rule 1If functions

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

moreover

those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.

Consequence. If two differentiable functions differ by a constant, then their derivatives are, i.e.

Rule 2If functions

are differentiable at some point , then their product is also differentiable at the same point

moreover

those. the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other.

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

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

For example, for three multipliers:

Rule 3If functions

differentiable at some point and , then at this point their quotient is also differentiable.u/v , and

those. the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator.

Where to look on other pages

When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so more examples on these derivatives are in the article."The derivative of a product and a quotient".

Comment. You should not confuse a constant (that is, a number) as a term in the sum and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. This is a typical mistake that occurs at the initial stage of studying derivatives, but as the average student solves several one-two-component examples, this mistake no longer makes.

And if, when differentiating a product or a quotient, you have a term u"v, wherein u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (such a case is analyzed in example 10).

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

Along the way, you can not do without transformations of expressions. To do this, you may need to open in new windows manuals Actions with powers and roots and Actions with fractions .

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

If you have a task like , then you are in the lesson "Derivatives of simple trigonometric functions".

Step by step examples - how to find the derivative

Example 3 Find the derivative of a function

Solution. We determine the parts of the expression of the function: the entire expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the product differentiation rule: the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other:

Next, we apply the rule of differentiation of the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum, we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values ​​of derivatives:

We substitute the found derivatives into the sum of products and obtain the derivative of the entire function required by the condition of the problem:

Example 4 Find the derivative of a function

Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating a quotient: the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator. We get:

We have already found the derivative of the factors in the numerator in Example 2. Let's also not forget that the product, which is the second factor in the numerator in the current example, is taken with a minus sign:

If you are looking for solutions to such problems in which you need to find the derivative of a function, where there is a continuous pile of roots and degrees, such as, for example, then welcome to class "The derivative of the sum of fractions with powers and roots" .

If you need to learn more about the derivatives of sines, cosines, tangents and other trigonometric functions, that is, when the function looks like , then you have a lesson "Derivatives of simple trigonometric functions" .

Example 5 Find the derivative of a function

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

Example 6 Find the derivative of a function

Solution. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the tabular value of the derivative of the square root, we obtain.