EntranceWhat is the derivative of arcsin 1 sqrt 3. Derivative of e to the x power and the exponential function
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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 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 an irrational number, in the fourth case we have the derivative of zero (zero is an integer), in the fifth - the 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.
Derivative formula power function looks like 
, 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, we 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 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 functions 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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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 elementary functions and apply the basic rules of differentiation:
- Moving the constant beyond the sign of the derivative: $$ (Cu)" = C(u)" $$
- Derivative of the sum/difference of functions: $$ (u \pm v)" = (u)" \pm (v)" $$
- Derivative of the product of two functions: $$ (u \cdot v)" = u"v + uv" $$
- Derivative of a fraction: $$ \bigg (\frac(u)(v) \bigg)" = \frac(u"v - uv"))(v^2) $$
- Derivative of a complex function: $$ (f(g(x)))" = f"(g(x)) \cdot g"(x) $$
Examples of solutions
| Example 1 |
| Find the derivative of the function $ y = x^3 - 2x^2 + 7x - 1 $ |
| Solution |
|
The derivative of the sum/difference of functions is equal to the sum/difference of derivatives: $$ y" = (x^3 - 2x^2 + 7x - 1)" = (x^3)" - (2x^2)" + (7x)" - (1)" = $$ Using the rule for the derivative of a power function $ (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 a constant is equal to zero. If you cannot solve your problem, then send it to us. We will provide detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner! |
| Answer |
| $$y" = 3x^2 - 4x + 7 $$ |
The problem of finding the derivative of a given function is one of the main ones in high school mathematics courses and in higher education. 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 denote them we will use 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.
- (sin x)"=cos x
- (cos x)"= –sin x
- (x n)"=n x n-1
- (e x)"=e x
- (ln x)"=1/x
- (a x)"=a x ln a
- (log a x)"=1/x ln a
- (tg x)"=1/cos 2 x
- (ctg x)"= – 1/sin 2 x
- (arcsin x)"= 1/√(1-x 2)
- (arccos x)"= - 1/√(1-x 2)
- (arctg x)"= 1/(1+x 2)
- (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 is an 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 it out constant coefficient 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 an external 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.
Date: 11/20/2014
What is a derivative?
Table of derivatives.
Derivative is one of the main concepts 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 variety, there are functions that are most important for practical application. 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 the power function in general view(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 - 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.
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.