How to find the derivative?

Rules of differentiation.

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 derivatives in general). What a derivative is and how to work with the table of derivatives is clearly explained in the previous lesson. Here we will deal with the rules of differentiation.

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

Expression "rules of differentiation" refers to finding the derivative from arithmetic operations. This understanding helps a lot to avoid confusion in your 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 get shortchanged.) It’s just that rule 4 is an elementary consequence of rule 3. But it is so popular that it makes sense to write (and remember!) it as an independent formula.

Under the designations 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 the function V. We have every right to write:

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

That’s better, right?) All that remains is to find the derivatives of sine and square of x. There is a table of derivatives for this. We just look for the functions we need in the table ( sinx And x 2), look at what derivatives they have and write down the answer:

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

That's it. Rule 1 of sum differentiation works exactly the same.

What if we have several terms? No problem.) We break the function into terms and look for the derivative of each term independently of the others. For example:

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

We boldly write:

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

At the end of the lesson I will give tips to make life easier when differentiating.)

Practical advice:

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

2. In complicated examples, we describe the solution in detail, with all the parentheses and dashes.

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

The problem of finding the derivative of a given function is one of the main ones in high school mathematics courses and in higher educational institutions. It is impossible to fully explore a function and construct 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 basic functions. Let's figure out how to find the derivative of a function.

The derivative of a function is 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.

Understanding this definition is quite difficult, since the concept of a limit is not fully studied in 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 move straight to finding the derivative.

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

To designate them we will use the Latin letters f, g, etc.

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

Derivatives 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. (sin x)"=cos x
2. (cos x)"= –sin x
3. (x n)"=n x n-1
4. (e x)"=e x
5. (ln x)"=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 this is a constant. From the table of derivatives it is known that the derivative of a 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 reduce it by 1 (formula 3).

(x 100)"=100 x 99

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

This exponential function, let's 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 ln 4

Rules of differentiation

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

1. The 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 a constant factor of 6 and differentiate only x 4. This is a power function, the derivative of which is found using formula 3 of the table of derivatives.

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

2. The derivative of a 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

A function is the sum of two functions, the derivatives of which 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, the derivatives of which we can also find in the table. Then the derivative of the difference is equal to the difference of the derivatives and don’t 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; let’s 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 equal to:

(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, we 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. We 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 a quotient, we first find the derivative of the numerator and denominator separately: (x 50)"=50 x 49 and (sin x)"= cos x. Substituting the derivative of the quotient into the formula, 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. There is also a rule for finding the derivative of a complex function:

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

Let's figure out how to find the derivative of such a function. Let y= u(v(x)) be a complex function. Let's call the function u 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, you need to multiply the derivatives of the internal and external functions.

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

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

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

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 over 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.

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. Online calculator It will also be useful in the case when there is a need to check the correctness of your solution, and if it is incorrect, quickly find the error.

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The derivative is constant.

When deriving the very first formula of the table, we will proceed from the definition of the derivative of a function at a point. Let's take , where x is any real number, that is, x is 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 , 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, the derivative of a constant function is equal to zero throughout the entire domain of definition.

Example.

Find derivatives of the following constant functions

Solution.

In the first case we have the derivative of the natural number 3, in the second case we have to take the derivative of the parameter a, which can be any real number, in the third - the derivative of the irrational number, in the fourth case we have the derivative of zero (zero is an integer), in the fifth – derivative of a rational fraction.

Answer:

The derivatives of all these functions are equal to zero for any real x (over 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 is 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, let's turn to the formula:

Hence,

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

Two cases should be considered: for positive x and negative x.

Let's assume first. In this case . Let's take the logarithm of the equality to base e and apply the property of the logarithm:

We arrived at an implicitly specified function. We find its derivative:

It remains to carry out the proof for negative x.

When the exponent p is an even number, then the power function is also defined for and is even (see section). That is, . In this case, you can also use the proof through the logarithmic derivative.

When the exponent p is an odd number, then the power function is also defined for and is odd. That is, . In this case, the logarithmic derivative cannot be used. To prove the formula in this case, you can use the rules of differentiation and the rule for finding the derivative of a complex function:

The last transition is possible due to the fact that if p is an odd number, then p-1 is either an even number or zero (for p=1), therefore, for negative x the equality is true .

Thus, the formula for the derivative of a power function is proven for any real p.

Example.

Find derivatives of functions.

Solution.

We bring the first and third functions to a tabular form using the properties of a power, and apply the formula for the derivative of a power function:

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:

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 is cos x.

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

When solving differentiation problems, we will constantly refer to the table of derivatives of basic functions, otherwise why did we compile it and prove each formula. We recommend that you remember all these formulas; in the future it will save you a lot of time.

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