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 School Mathematics Formulas. The reference manual can be opened or downloaded from the page Mathematical formulas and tables. Also from there we need Derivative table, it is better to print it, you will often have to refer to it, and not only now, but also offline.
There is? Let's get started. I have two news for you: good and very good. The good news is this: in order to learn how to find derivatives, it is not at all necessary to know and understand what a derivative is. Moreover, the definition of the derivative of a function, the mathematical, physical, geometric meaning of the derivative is more expedient to digest later, since the qualitative 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 very 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 study of the topic A: First, this article. Then you need to read the most important lesson Derivative of a compound function. These two basic classes will allow you to raise your skills from scratch. Further, it will be possible to familiarize yourself with more complex derivatives in the article. complex derivatives. logarithmic derivative. If the bar is too high, read the item first The simplest typical problems with a derivative. In addition to the new material, the lesson covered other, simpler types of derivatives, and there is a great opportunity to improve your differentiation technique. In addition, in the control work, there are almost always tasks for finding derivatives of functions that are specified implicitly or parametrically. There is also a tutorial for this: 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 look at an example:
Example 1
Find the derivative of a function
Decision: 
This is the simplest 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 .
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 .
Notation: The derivative is denoted by or .
ATTENTION, IMPORTANT! Forget to put a stroke (where necessary), or draw an extra stroke (where it is not necessary) - BIG ERROR! 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 a constant:
, where is a constant number;
derivative of a power function:
, in particular: , , .
Why memorize? This knowledge is elementary knowledge about derivatives. And if you can’t answer the teacher’s question “What is the derivative of the number?”, Then your studies at the university may end for you (I personally know two real cases from life). In addition, these are the most common formulas that we have to use almost every time we encounter derivatives.
In reality, simple tabular examples are rare; usually, when finding derivatives, differentiation rules are used first, and then a table of derivatives of elementary functions.
In this regard, we turn to the consideration differentiation rules:
1) A constant number can (and should) be taken out of the sign of the derivative
Where is a constant number (constant)
Example 2
Find the derivative of a function
We 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 out the constant factor beyond the sign of the derivative:
And now we turn our cosine according to the table:
Well, it is desirable to “comb” the result a little - put the minus in the 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
We decide. As you probably already noticed, the first action that is always performed when finding the derivative is that we put the whole expression in brackets and put a dash on the top right:
We apply the second rule:
Please note that for differentiation, all roots, degrees must be represented as , and if they are in the denominator, then move them up. How to do this is discussed in my methodological materials.
Now we recall the first rule of differentiation - we take out the constant factors (numbers) outside the sign of the derivative:
Usually, during the solution, these two rules are applied simultaneously (so as not to rewrite a long expression once again).
All functions under the dashes are elementary table functions, using the table we perform the transformation:
You can leave everything in this form, since there are no more strokes, and the derivative has been found. However, expressions like this usually simplify:
It is desirable to represent all degrees of the species again as roots, and to reset the degrees with negative indicators to the denominator. Although you can not do this, it will not 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, by analogy, the formula suggests itself ...., but the surprise is that:
This unusual rule (as well as others) follows from definitions of the derivative. But we will wait with the theory for now - now it is 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 according to the table of derivatives:
Complicated? Not at all, quite affordable even for a teapot.
Example 6
Find the derivative of a function 
This function contains the sum and product of two functions - a square trinomial and a logarithm. We remember from school 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 have only elementary functions left, according to the table of derivatives we turn 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 resolved verbally, and it is immediately recorded that 
.
Example 7
Find the derivative of a function 
This is an example for self-solving (answer at the end of the lesson)
4) Derivative of private functions
A hatch has opened in the ceiling, don't be scared, it's a glitch.
And here is the harsh reality: 
Example 8
Find the derivative of a function
What is not here - the sum, difference, product, fraction .... Where to begin?! There are doubts, no doubts, but, ANYWAY first, draw brackets and put a stroke at the top right:
Now we look at the expression in brackets, how would we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to take it out of the sign of the derivative.
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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Derivation of the formula for the derivative of a power function (x to the power of a). Derivatives of roots from x are considered. The formula for the derivative of a higher order power function. Examples of calculating derivatives.
ContentSee also: Power function and roots, formulas and graph
Power Function Plots
Basic Formulas
The derivative of x to the power of a is a times x to the power of a minus one:
(1)
.
The derivative of the nth root of x to the mth power is:
(2)
.
Derivation of the formula for the derivative of a power function
Case x > 0
Consider a power function of variable x with exponent a :
(3)
.
Here a is an arbitrary real number. Let's consider the case first.
To find the derivative of the function (3), we use the properties of the power function and transform it to the following form:
.
Now we find the derivative by applying:
;
.
Here .
Formula (1) is proved.
Derivation of the formula for the derivative of the root of the degree n of x to the degree m
Now consider a function that is the root of the following form:
(4)
.
To find the derivative, we convert the root to a power function:
.
Comparing with formula (3), we see that
.
Then
.
By formula (1) we find the derivative:
(1)
;
;
(2)
.
In practice, there is no need to memorize formula (2). It is much more convenient to first convert the roots to power functions, and then find their derivatives using formula (1) (see examples at the end of the page).
Case x = 0
If , then the exponential function is also defined for the value of the variable x = 0
. Let us find the derivative of function (3) for x = 0
. To do this, we use the definition of a derivative:
.
Substitute x = 0
:
.
In this case, by derivative we mean the right-hand limit for which .
So we found:
.
From this it can be seen that at , .
At , .
At , .
This result is also obtained by formula (1):
(1)
.
Therefore, formula (1) is also valid for x = 0
.
case x< 0
Consider function (3) again:
(3)
.
For some values of the constant a , it is also defined for negative values of the variable x . Namely, let a be a rational number. Then it can be represented as an irreducible fraction:
,
where m and n are integers with no common divisor.
If n is odd, then the exponential function is also defined for negative values of the variable x. For example, for n = 3
and m = 1
we have the cube root of x :
.
It is also defined for negative values of x .
Let us find the derivative of the power function (3) for and for rational values of the constant a , for which it is defined. To do this, we represent x in the following form:
.
Then ,
.
We find the derivative by taking the constant out of the sign of the derivative and applying the rule of differentiation of a complex function:
.
Here . But
.
Because , then
.
Then
.
That is, formula (1) is also valid for:
(1)
.
Derivatives of higher orders
Now we find the higher order derivatives of the power function
(3)
.
We have already found the first order derivative:
.
Taking the constant a out of the sign of the derivative, we find the second-order derivative:
.
Similarly, we find derivatives of the third and fourth orders:
;
.
From here it is clear that derivative of an arbitrary nth order has the following form:
.
notice, that if a is a natural number, , then the nth derivative is constant:
.
Then all subsequent derivatives are equal to zero:
,
at .
Derivative Examples
Example
Find the derivative of the function:
.
Let's convert the roots to powers:
;
.
Then the original function takes the form:
.
We find derivatives of degrees:
;
.
The derivative of a constant is zero:
.
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:
- Taking the constant out of the sign of the derivative: $$ (Cu)" = C(u)" $$
- Derivative of sum/difference of functions: $$ (u \pm v)" = (u)" \pm (v)" $$
- Derivative of the product of two functions: $$ (u \cdot v)" = u"v + uv" $$
- Fraction derivative : $$ \bigg (\frac(u)(v) \bigg)" = \frac(u"v - uv")(v^2) $$
- Compound function derivative : $$ (f(g(x)))" = f"(g(x)) \cdot g"(x) $$
Solution examples
| Example 1 |
| Find the derivative of the function $ y = x^3 - 2x^2 + 7x - 1 $ |
| Decision |
|
The derivative of the sum/difference of functions is equal to the sum/difference of the derivatives: $$ y" = (x^3 - 2x^2 + 7x - 1)" = (x^3)" - (2x^2)" + (7x)" - (1)" = $$ Using the power function derivative rule $ (x^p)" = px^(p-1) $ we have: $$ y" = 3x^(3-1) - 2 \cdot 2 x^(2-1) + 7 - 0 = 3x^2 - 4x + 7 $$ It was also taken into account that the derivative of the constant is equal to zero. If you can't solve your problem, then send her to us. We will provide a detailed solution. You will be able to familiarize yourself with the progress of the calculation and gather information. This will help you get a credit from the teacher in a timely manner! |
| Answer |
| $$ y" = 3x^2 - 4x + 7 $$ |