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. An online calculator will also be useful in cases where there is a need to check whether your solution is correct, and if it is incorrect, quickly find an error.
Determining the derivative of a function is the inverse operation of integrating a function. For elementary functions, calculating the derivative is not difficult; just use the table of derivatives. If we need find the derivative from a complex function, then differentiation will be much more difficult and will require more care and time. At the same time, it is very easy to make a typo or a minor mistake that will lead to a final incorrect answer. Therefore, it is always important to be able to check your decision. You can do this using this online calculator, which allows you to find derivatives of any functions online with a detailed solution for free, without registering on the site. Finding the derivative of a function (differentiation) is the ratio of the increment of the function to the increment of the argument (numerically, the derivative is equal to the tangent of the tangent to the graph of the function). If you need to calculate the derivative of a function at a specific point, then you need in the received answer instead of an argument x substitute its numerical value and calculate the expression. At online derivative solution you need to enter the function in the appropriate field: the argument must be a variable x, since differentiation occurs precisely along it. To calculate the second derivative, you need to differentiate the resulting answer.
Date: 11/20/2014
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 set there are functions that are most important for practical use. 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 a power function in general form (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 very 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.
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 of 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 a quotient in real problems, it is always necessary to apply several differentiation rules at once, so there are more examples on 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 is a typical mistake that 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).
Another common mistake is mechanically solving the derivative of a complex function as the 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:
And you can check the solution to the derivative problem on.
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 us 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:
You can check the solution to the derivative problem at online derivatives calculator .
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 .
Very easy to remember.
Well, let’s not go far, let’s immediately consider the inverse function. Which function is the inverse of the exponential function? Logarithm:
In our case, the base is the number:
Such a logarithm (that is, a logarithm with a base) is called “natural”, and we use a special notation for it: we write instead.
What is it equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find the derivative of the function.
- What is the derivative of the function?
Answers: The exponential and natural logarithm are uniquely simple functions from a derivative perspective. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.
Rules of differentiation
Rules of what? Again a new term, again?!...
Differentiation is the process of finding the derivative.
That's all. What else can you call this process in one word? Not derivative... Mathematicians call the differential the same increment of a function at. This term comes from the Latin differentia - difference. Here.
When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
There are 5 rules in total.
The constant is taken out of the derivative sign.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let it be, or simpler.
Examples.
Find the derivatives of the functions:
- at a point;
- at a point;
- at a point;
- at the point.
Solutions:
- (the derivative is the same at all points, since it is a linear function, remember?);
Derivative of the product
Everything is similar here: let’s introduce a new function and find its increment:
Derivative:
Examples:
- Find the derivatives of the functions and;
- Find the derivative of the function at a point.
Solutions:
Derivative of an exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just exponents (have you forgotten what that is yet?).
So, where is some number.
We already know the derivative of the function, so let's try to reduce our function to a new base:
To do this, we will use a simple rule: . Then:
Well, it worked. Now try to find the derivative, and don't forget that this function is complex.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative of an exponent: as it was, it remains the same, only a factor appeared, which is just a number, but not a variable.
Examples:
Find the derivatives of the functions:
Answers:
This is just a number that cannot be calculated without a calculator, that is, it cannot be written down in a simpler form. Therefore, we leave it in this form in the answer.
Note that here is the quotient of two functions, so we apply the corresponding differentiation rule:
In this example, the product of two functions:
Derivative of a logarithmic function
It’s similar here: you already know the derivative of the natural logarithm:
Therefore, to find an arbitrary logarithm with a different base, for example:
We need to reduce this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now we will write instead:
The denominator is simply a constant (a constant number, without a variable). The derivative is obtained very simply:
Derivatives of exponential and logarithmic functions are almost never found in the Unified State Examination, but it will not be superfluous to know them.
Derivative of a complex function.
What is a "complex function"? No, this is not a logarithm, and not an arctangent. These functions can be difficult to understand (although if you find the logarithm difficult, read the topic “Logarithms” and you will be fine), but from a mathematical point of view, the word “complex” does not mean “difficult”.
Imagine a small conveyor belt: two people are sitting and doing some actions with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second one ties it with a ribbon. The result is a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the reverse steps in reverse order.
Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then square the resulting number. So, we are given a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we perform the first action directly with the variable, and then a second action with what resulted from the first.
In other words, a complex function is a function whose argument is another function: .
For our example, .
We can easily do the same steps in reverse order: first you square it, and I then look for the cosine of the resulting number: . It’s easy to guess that the result will almost always be different. An important feature of complex functions: when the order of actions changes, the function changes.
Second example: (same thing). .
The action we do last will be called "external" function, and the action performed first - accordingly "internal" function(these are informal names, I use them only to explain the material in simple language).
Try to determine for yourself which function is external and which internal:
Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function
- What action will we perform first? First, let's calculate the sine, and only then cube it. This means that it is an internal function, but an external one.
And the original function is their composition: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: .
We change variables and get a function.
Well, now we will extract our chocolate bar and look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. In relation to the original example, it looks like this:
Another example:
So, let's finally formulate the official rule:
Algorithm for finding the derivative of a complex function:
It seems simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(Just don’t try to cut it by now! Nothing comes out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that this is a three-level complex function: after all, this is already a complex function in itself, and we also extract the root from it, that is, we perform the third action (put the chocolate in a wrapper and with a ribbon in the briefcase). But there is no reason to be afraid: we will still “unpack” this function in the same order as usual: from the end.
That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.
In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:
The later the action is performed, the more “external” the corresponding function will be. The sequence of actions is the same as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sine. .
4. Square. .
5. Putting it all together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN THINGS
Derivative of a function- the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument:
Basic derivatives:
Rules of differentiation:
The constant is taken out of the derivative sign:
Derivative of the sum:
Derivative of the product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the “internal” function and find its derivative.
- We define the “external” function and find its derivative.
- We multiply the results of the first and second points.
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