The proof and derivation of the formula for the cosine derivative - cos(x) is presented. Examples of calculating derivatives of cos 2x, cos 3x, cos nx, cosine squared, cubed and to the power of n. The formula for the derivative of the cosine of the nth order.
ContentSee also: Sine and cosine - properties, graphs, formulas
The derivative with respect to the variable x of the cosine of x is equal to minus the sine of x:
(cos x)′ = - sin x.
Proof
To derive the formula for the cosine derivative, we use the definition of the derivative:
.
Let's transform this expression to reduce it to known mathematical laws and rules. To do this, we need to know four properties.
1)
Trigonometric Formulas. We need the following formula:
(1)
;
2)
Continuity property of the sine function:
(2)
;
3)
The meaning of the first remarkable limit:
(3)
;
4)
The limit property of the product of two functions:
If and then
(4)
.
We apply these laws to our limit. First we transform the algebraic expression
.
For this we apply the formula
(1)
;
In our case
; . Then
;
;
;
.
Let's make a substitution. At , . We use the continuity property (2):
.
We make the same substitution and apply the first remarkable limit (3):
.
Since the limits calculated above exist, we apply property (4):
.
Thus, we have obtained the formula for the derivative of the cosine.
Examples
Consider simple examples of finding derivatives of functions containing cosine. Let's find the derivatives of the following functions:
y = cos2x; y = cos 3x; y = cos nx; y= cos 2 x; y= cos 3 x and y= cos n x.
Example 1
Find derivatives of cos 2x, cos 3x and cos nx.
The original functions have a similar form. Therefore, we will find the derivative of the function y = cos nx. Then, as a derivative of cos nx, substitute n = 2 and n = 3 . And, thus, we obtain formulas for derivatives of cos 2x and cos 3x .
So, we find the derivative of the function
y = cos nx
.
Let's represent this function of variable x as a complex function consisting of two functions:
1)
2)
Then the original function is a complex (composite) function composed of the functions and :
.
Let's find the derivative of the function with respect to the variable x:
.
Let's find the derivative of the function with respect to the variable:
.
We apply.
.
Substitute :
(P1) .
Now, in the formula (P1) we substitute and :
;
.
;
;
.
Example 2
Find derivatives of cosine squared, cosine cubed, and cosine raised to the power of n:
y= cos 2 x; y= cos 3 x; y= cos n x.
In this example, the functions also have a similar look. Therefore, we will find the derivative of the most general function - the cosine to the power of n:
y= cos n x.
Then we substitute n = 2 and n = 3 . And, thus, we obtain formulas for the derivatives of cosine squared and cosine cubed.
So, we need to find the derivative of the function
.
Let's rewrite it in a more understandable form:
.
Let's represent this function as a complex function consisting of two functions:
1)
Variable dependent functions : ;
2)
Variable dependent functions : .
Then the original function is a complex function composed of two functions and :
.
We find the derivative of the function with respect to the variable x:
.
We find the derivative of the function with respect to the variable:
.
We apply the rule of differentiation of a complex function.
.
Substitute :
(P2) .
Now let's substitute and:
;
.
;
;
.
Derivatives of higher orders
Note that the derivative of cos x of the first order can be expressed in terms of cosine as follows:
.
Let's find the second order derivative using the formula for the derivative of a complex function:
.
Here .
Note that differentiation cos x causes its argument to be incremented by . Then the derivative of the nth order has the form:
(5)
.
This formula can be proved more strictly using the method of mathematical induction. The proof for the nth derivative of the sine is given on the page “Derivative of the sine”. For the nth derivative of the cosine, the proof is exactly the same. It is only necessary to replace sin with cos in all formulas.
See also: Derivative calculation is one of the most important operations in differential calculus. Below is a table for finding derivatives of simple functions. For more complex differentiation rules, see other lessons:- Table of derivatives of exponential and logarithmic functions
Derivatives of simple functions
1. The derivative of a number is zeroс´ = 0
Example:
5' = 0
Explanation:
The derivative shows the rate at which the value of the function changes when the argument changes. Since the number does not change in any way under any conditions, the rate of its change is always zero.
2. Derivative of a variable equal to one
x' = 1
Explanation:
With each increment of the argument (x) by one, the value of the function (calculation result) increases by the same amount. Thus, the rate of change of the value of the function y = x is exactly equal to the rate of change of the value of the argument.
3. The derivative of a variable and a factor is equal to this factor
сx´ = с
Example:
(3x)´ = 3
(2x)´ = 2
Explanation:
In this case, each time the function argument ( X) its value (y) grows in With once. Thus, the rate of change of the value of the function with respect to the rate of change of the argument is exactly equal to the value With.
Whence it follows that
(cx + b)" = c
that is, the differential of the linear function y=kx+b is equal to the slope of the straight line (k).
4. Modulo derivative of a variable is equal to the quotient of this variable to its modulus
|x|"= x / |x| provided that x ≠ 0
Explanation:
Since the derivative of the variable (see formula 2) is equal to one, the derivative of the modulus differs only in that the value of the rate of change of the function changes to the opposite when crossing the origin point (try to draw a graph of the function y = |x| and see for yourself. This is exactly value and returns the expression x / |x| When x< 0 оно равно (-1), а когда x >0 - one. That is, with negative values of the variable x, with each increase in the change in the argument, the value of the function decreases by exactly the same value, and with positive values, on the contrary, it increases, but by exactly the same value.
5. Power derivative of a variable is equal to the product of the number of this power and the variable in the power, reduced by one
(x c)"= cx c-1, provided that x c and cx c-1 are defined and c ≠ 0
Example:
(x 2)" = 2x
(x 3)" = 3x 2
To memorize the formula:
Take the exponent of the variable "down" as a multiplier, and then decrease the exponent itself by one. For example, for x 2 - two was ahead of x, and then the reduced power (2-1 = 1) just gave us 2x. The same thing happened for x 3 - we lower the triple, reduce it by one, and instead of a cube we have a square, that is, 3x 2 . A little "unscientific", but very easy to remember.
6.Fraction derivative 1/x
(1/x)" = - 1 / x 2
Example:
Since a fraction can be represented as raising to a negative power
(1/x)" = (x -1)" , then you can apply the formula from rule 5 of the derivatives table
(x -1)" = -1x -2 = - 1 / x 2
7. Fraction derivative with a variable of arbitrary degree in the denominator
(1/x c)" = - c / x c+1
Example:
(1 / x 2)" = - 2 / x 3
8. root derivative(derivative of variable under square root)
(√x)" = 1 / (2√x) or 1/2 x -1/2
Example:
(√x)" = (x 1/2)" so you can apply the formula from rule 5
(x 1/2)" \u003d 1/2 x -1/2 \u003d 1 / (2√x)
9. Derivative of a variable under a root of an arbitrary degree
(n √ x)" = 1 / (n n √ x n-1)
The calculation of the derivative is often found in USE assignments. This page contains a list of formulas for finding derivatives.
Differentiation rules
- (k⋅f(x))′=k⋅f′(x).
- (f(x)+g(x))′=f′(x)+g′(x).
- (f(x)⋅ g(x))′=f′(x)⋅ g(x)+f(x)⋅ g′(x).
- Derivative of a complex function. If y=F(u) and u=u(x), then the function y=f(x)=F(u(x)) is called a complex function of x. Is equal to y′(x)=Fu′⋅ ux′.
- Derivative of an implicit function. The function y=f(x) is called the implicit function given by the relation F(x,y)=0 if F(x,f(x))≡0.
- Derivative of the inverse function. If g(f(x))=x, then the function g(x) is called the inverse function for the function y=f(x).
- Derivative of a parametrically given function. Let x and y be given as functions of the variable t: x=x(t), y=y(t). It is said that y=y(x) is a parametrically defined function on the interval x∈ (a;b) if on this interval the equation x=x(t) can be expressed as t=t(x) and the function y=y( t(x))=y(x).
- Derivative of exponential function. It is found by taking the logarithm to the base of the natural logarithm.
The proof and derivation of the formula for the derivative of the sine - sin(x) is presented. Examples of calculating derivatives of sin 2x, sine squared and cubed. Derivation of the formula for the derivative of the sine of the nth order.
ContentSee also: Sine and cosine - properties, graphs, formulas
The derivative with respect to the variable x of the sine of x is equal to the cosine of x:
(sin x)′ = cos x.
Proof
To derive the formula for the derivative of the sine, we will use the definition of the derivative:
.
To find this limit, we need to transform the expression in such a way as to reduce it to known laws, properties, and rules. To do this, we need to know four properties.
1)
Meaning of the first remarkable limit:
(1)
;
2)
Continuity of the cosine function:
(2)
;
3)
Trigonometric Formulas. We need the following formula:
(3)
;
4)
Arithmetic properties of the function limit:
If and then
(4)
.
We apply these rules to our limit. First we transform the algebraic expression
.
For this we apply the formula
(3)
.
In our case
; . Then
;
;
;
.
Now let's make a substitution. At , . Let us apply the first remarkable limit (1):
.
We make the same substitution and use the continuity property (2):
.
Since the limits calculated above exist, we apply property (4):
.
The formula for the derivative of the sine has been proven.
Examples
Consider simple examples of finding derivatives of functions containing a sine. We will find derivatives of the following functions:
y=sin2x; y= sin2x and y= sin3x.
Example 1
Find the derivative of sin 2x.
First we find the derivative of the simplest part:
(2x)′ = 2(x)′ = 2 1 = 2.
We apply.
.
Here .
(sin 2x)′ = 2 cos 2x.
Example 2
Find the derivative of the squared sine:
y= sin2x.
Let's rewrite the original function in a more understandable form:
.
Find the derivative of the simplest part:
.
We apply the formula for the derivative of a complex function.
.
Here .
One of the trigonometry formulas can be applied. Then
.
Example 3
Find the derivative of the sine cubed:
y= sin3x.
Derivatives of higher orders
Note that the derivative of sin x of the first order can be expressed in terms of the sine as follows:
.
Let's find the second order derivative using the formula for the derivative of a complex function:
.
Here .
Now we can see that the differentiation sin x causes its argument to be incremented by . Then the derivative of the nth order has the form:
(5)
.
Let us prove this by applying the method of mathematical induction.
We have already checked that for , formula (5) is valid.
Let us assume that formula (5) is valid for some value of . Let us prove that it follows from this that formula (5) is valid for .
We write formula (5) for :
.
We differentiate this equation by applying the rule of differentiation of a complex function:
.
Here .
So we found:
.
If we substitute , then this formula takes the form (5).
The formula has been proven.
See also:Date: 11/20/2014
What is a derivative?
Derivative table.
The derivative is one of the main concepts of higher mathematics. In this lesson, we will introduce this concept. Let's get acquainted, without strict mathematical formulations and proofs.
This introduction will allow you to:
Understand the essence of simple tasks with a derivative;
Successfully solve these very simple tasks;
Prepare for more serious derivative lessons.
First, a pleasant surprise.
The strict definition of the derivative is based on the theory of limits, and the thing is rather complicated. It's upsetting. But the practical application of the derivative, 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. And that's it. This makes me happy.
Shall we get to know each other?)
Terms and designations.
There are many mathematical operations in elementary mathematics. Addition, subtraction, multiplication, exponentiation, logarithm, etc. If one more operation is added 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.
Here it is important to understand that differentiation is just a mathematical operation on a function. We take any function and, according to certain rules, transform it. The result is a new function. This new function is called: derivative.
Differentiation- action on a function.
Derivative is the result of this action.
Just like, for example, sum is the result of the addition. Or private is the result of the division.
Knowing the terms, you can at least understand the tasks.) The wording is as follows: find the derivative of a function; take the derivative; differentiate the function; calculate derivative etc. It's all same. Of course, there are more complex tasks, where finding the derivative (differentiation) will be just one of the steps in solving the task.
The derivative is denoted by a dash at the top right above the function. Like this: y" or f"(x) or S"(t) and so on.
read y stroke, ef stroke from x, es stroke from te, well you get it...)
A prime can also denote the derivative of a particular function, for example: (2x+3)", (x 3 )" , (sinx)" etc. Often the derivative is denoted using differentials, but we will not consider such a notation in this lesson.
Suppose that we have learned to understand the tasks. There is nothing left - to learn how to solve them.) Let me remind you again: finding the derivative is transformation of a function according to certain rules. These rules are surprisingly few.
To find the derivative of a function, you only need to know three things. Three pillars on which all differentiation rests. Here are the three whales:
1. Table of derivatives (differentiation formulas).
3. Derivative of a complex function.
Let's start in order. In this lesson, we will consider the table of derivatives.
Derivative table.
The world has an infinite number of functions. Among this set there are functions which are most important for practical application. These functions sit in all the laws of nature. From these functions, as 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 the derivative and the theory of limits - a rather time-consuming thing. And mathematicians are people too, yes, yes!) So they simplified their lives (and us). They calculated 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. Left - elementary function, right - its derivative.
Function y |
Derivative of function y y" |
|
1 | C (constant) | C" = 0 |
2 | x | x" = 1 |
3 | x n (n is any number) | (x n)" = nx n-1 |
x 2 (n = 2) | (x 2)" = 2x | |
4 | sin x | (sinx)" = cosx |
cos x | (cos x)" = - sin x | |
tg x | ||
ctg x | ||
5 | arcsin x | |
arccos x | ||
arctg 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! Is the hint clear?) Yes, it is desirable 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 tabular 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 formulation of the task, or in the original function, which does not 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 general derivative of the power function (third group). In our case, n=3. So we substitute the triple instead of n and carefully write down the result:
(x 3) " = 3 x 3-1 = 3x 2
That's all there is to 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 to this same derivative. It's 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 already a new function.
On the plate we find the sine and the corresponding derivative:
y" = (sinx)" = cosx
Substitute zero into the derivative:
y"(0) = cos 0 = 1
This will be the answer.
3. Differentiate the function:
What inspires?) There is not even close such a 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, finding the derivative of our function is quite troublesome. The table doesn't help...
But if we see that our function is cosine of a double angle, then everything immediately gets better!
Yes Yes! Remember that the transformation of the original function before differentiation quite acceptable! And it happens to make life a lot easier. According to the formula for the cosine of a double angle:
Those. our tricky function is nothing but y = cox. And this is a table function. We immediately get:
Answer: y" = - sin x.
Example for advanced graduates and students:
4. Find the derivative of a function:
There is no such function in the derivatives table, of course. But if you remember elementary mathematics, actions 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! The third group, n=1/10. Directly according to the formula and write:
That's all. This will be the answer.
I hope that with the first whale of differentiation - the table of derivatives - everything is clear. It remains to deal with the two remaining whales. In the next lesson, we will learn the rules of differentiation.