Determining the derivative of a function is the inverse operation of integrating a function. For elementary functions it is not difficult to calculate the derivative, it is enough to use the table of derivatives. If we need find the derivative from complex function, then differentiation will be much more difficult, it will require more care and time. It is very easy to make a typo or a minor mistake, which will lead to the final wrong answer. Therefore, it is always important to be able to check your decision. You can do this with this online calculator, which allows you to find derivatives of any function online with detailed solution free, without registration 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 slope of the tangent to the graph of the function). If it is necessary to calculate the derivative of a function at a specific point, then in the received response, instead of the argument x frame him numerical value and calculate the expression. At online derivative solution you need to enter a function in the corresponding field: in this case, the argument must be a variable x, since differentiation goes exactly along it. To calculate the second derivative, you need to differentiate the received answer.
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
Decision. 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
Decision. Differentiate as a derivative of the sum, in which the second term with a constant factor, it 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 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 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
and
those. the derivative of the algebraic sum of functions is algebraic sum 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
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.
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 typical mistake, which occurs at the initial stage of the study of derivatives, but as the solution of several one-two-part examples, the average student no longer makes this mistake.
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. So 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
Decision. We determine the parts of the function expression: 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:
And you can check the solution of the problem on the derivative on .
Example 4 Find the derivative of a function
Decision. 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 derivatives of sines, cosines, tangents and others 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
Decision. 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:
You can check the solution of the derivative problem on derivative calculator online .
Example 6 Find the derivative of a function
Decision. 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 get:
To get rid of the fraction in the numerator, multiply the numerator and denominator by .
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.
It's 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 focus and remember everything, everything, everything 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 x2), 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.
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 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.
physical meaning 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 for some 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:
Decision: 
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.
For any questions on this and other topics, please contact 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.
