Rules for calculating derivatives. Derivative of the function Proper derivatives for the function f x
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 $ |
| Solution |
|
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 cannot solve your problem, then send it 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 |
| $$ y" = 3x^2 - 4x + 7 $$ |
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.
Find derivative online
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In this lesson, we will learn how to apply formulas and rules of differentiation.
Examples. Find derivatives of functions.
1. y=x 7 +x 5 -x 4 +x 3 -x 2 +x-9. Applying the Rule I, formulas 4, 2 and 1. We get:
y'=7x 6 +5x 4 -4x 3 +3x 2 -2x+1.
2. y=3x6 -2x+5. We solve similarly, using the same formulas and the formula 3.
y’=3∙6x 5 -2=18x 5 -2.
Applying the Rule I, formulas 3, 5
and 6
and 1.
Applying the Rule IV, formulas 5
and 1
.
In the fifth example, according to the rule I the derivative of the sum is equal to the sum of the derivatives, and we just found the derivative of the 1st term (example 4 ), therefore, we will find derivatives 2nd and 3rd terms, and for 1st term, we can immediately write the result.
Differentiating 2nd and 3rd terms according to the formula 4
. To do this, we transform the roots of the third and fourth degrees in denominators to powers with negative exponents, and then, according to 4
formula, we find the derivatives of the powers.
Look at this example and the result. Did you catch the pattern? Good. This means that we have a new formula and can add it to our derivatives table.
Let's solve the sixth example and derive one more formula.
We use the rule IV and formula 4
. We reduce the resulting fractions.
We look at this function and its derivative. You, of course, understood the pattern and are ready to name the formula:
Learning new formulas!
Examples.
1. Find argument increment and function increment y= x2 if the initial value of the argument was 4 , and the new 4,01 .
Solution.
New argument value x \u003d x 0 + Δx. Substitute the data: 4.01=4+Δx, hence the increment of the argument Δх=4.01-4=0.01. The increment of a function, by definition, is equal to the difference between the new and previous values of the function, i.e. Δy \u003d f (x 0 + Δx) - f (x 0). Since we have a function y=x2, then Δу\u003d (x 0 + Δx) 2 - (x 0) 2 \u003d (x 0) 2 + 2x 0 · Δx+(Δx) 2 - (x 0) 2 \u003d 2x 0 · ∆x+(∆x) 2 =
2 · 4 · 0,01+(0,01) 2 =0,08+0,0001=0,0801.
Answer: argument increment Δх=0.01; function increment Δу=0,0801.
It was possible to find the function increment in another way: Δy\u003d y (x 0 + Δx) -y (x 0) \u003d y (4.01) -y (4) \u003d 4.01 2 -4 2 \u003d 16.0801-16 \u003d 0.0801.
2. Find the angle of inclination of the tangent to the function graph y=f(x) at the point x 0, if f "(x 0) \u003d 1.
Solution.
The value of the derivative at the point of contact x 0 and is the value of the tangent of the slope of the tangent (the geometric meaning of the derivative). We have: f "(x 0) \u003d tgα \u003d 1 → α \u003d 45 °, because tg45°=1.
Answer: the tangent to the graph of this function forms an angle with the positive direction of the Ox axis, equal to 45°.
3. Derive the formula for the derivative of a function y=xn.
Differentiation is the act of finding the derivative of a function.
When finding derivatives, formulas are used that were derived on the basis of the definition of the derivative, in the same way as we derived the formula for the derivative degree: (x n)" = nx n-1.
Here are the formulas.
Derivative table it will be easier to memorize by pronouncing verbal formulations:
1. The derivative of a constant value is zero.
2. X stroke is equal to one.
3. The constant factor can be taken out of the sign of the derivative.
4. The derivative of a degree is equal to the product of the exponent of this degree by the degree with the same base, but the exponent is one less.
5. The derivative of the root is equal to one divided by two of the same roots.
6. The derivative of unity divided by x is minus one divided by x squared.
7. The derivative of the sine is equal to the cosine.
8. The derivative of cosine is equal to minus sine.
9. The derivative of the tangent is equal to one divided by the square of the cosine.
10. The derivative of the cotangent is minus one divided by the square of the sine.
We teach differentiation rules.
1.
The derivative of the algebraic sum is equal to the algebraic sum of the derivative terms.
2. The derivative of the product is equal to the product of the derivative of the first factor by the second plus the product of the first factor by the derivative of the second.
3. The derivative of “y” divided by “ve” is equal to a fraction, in the numerator of which “y is a stroke multiplied by “ve” minus “y, multiplied by a stroke”, and in the denominator - “ve squared”.
4. A special case of the formula 3.
Let's learn together!
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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)
