• How to find a derivative?

    The task of finding the derivative of a given function is one of the main ones in the course of high school mathematics and in higher educational institutions. It is impossible to fully explore the function, build its graph without taking its derivative. The derivative of a function can be easily found, knowing the basic rules of differentiation, as well as a table of derivatives of the basic functions. Let's see how to find the derivative of a function.

    The derivative of a function is the limit of the ratio of the increment of a function to the increment of an argument when the increment of the argument tends to zero.

    It is rather difficult to understand this definition, since the concept of the limit is not fully studied in school. But in order to find the derivatives of various functions, it is not necessary to understand the definition, let us leave it to the specialists of mathematicians and go straight to finding the derivative.

    The process of finding the derivative is called differentiation. When differentiating a function, we will get a new function.

    For their designation we will use the Latin letters f, g, etc.

    There are many possible notation for derivatives.We will use the stroke. For example, the entry g 'means that we will find the derivative of the function g.

    Derivatives Table

    In order to answer the question of how to find a 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. Simply look at its value in the derivative table.

    1. C '= 0
    2. (sin x) '= cos x
    3. (cos x) '= �sin x
    4. (xn) '= n xn-1
    5. (ex) '= ex
    6. (ln x) '= 1 / x
    7. (ax) '= axln a
    8. (logax) '= 1 / x ln a
    9. (tg x) '= 1 / cos2x
    10. (ctg x) '= - 1 / sin2x
    11. (arcsin x) '= 1 / √ (1-x2)
    12. (arccos x) '= - 1 / √ (1-x2)
    13. (arctg x) '= 1 / (1 + x2)
    14. (arcctg x) '= - 1 / (1 + x2)

    Example 1. Find the derivative of the function y = 500.

    We see that this is a constant. It is known from the derivative table that the derivative of a constant is zero (formula 1).

    (500)' = 0

    Example 2. Find the derivative of the function y = x100.

    This is a power function in the index of which 100 and in order to find its derivative, multiply the function by the exponent and decrease by 1 (formula 3).

    (x100) '= 100 x99

    Example 3. Find the derivative of the function y = 5x

    This is an exponential function, we calculate its derivative by formula 4.

    (5x)'= 5xln5

    Example 4. Find the derivative of the function y = log4x

    The derivative of the logarithm will find the formula 7.

    (log4x) '= 1 / x ln 4

    Differentiation rules

    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 also multiplication by a number). To find their derivatives, it is necessary to know the rules of differentiation. Further, 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

    (C f) '= C f'

    Example 5. Find the derivative of the function y = 6 * x8

    We take out the constant coefficient 6 and differentiate only x4. This is a power function, the derivative of which we find by the formula 3 of the table of derivatives.

    (6 * x8) '= 6 * (x8) '= 6 * 8 * x7= 48 * x7

    2. The derivative of the sum is equal to the sum of the derivatives.

    Then:

    (f + g) '= f' + g '

    Example 6. Find the derivative of the function y = x100+ sin x

    A function is the sum of two functions whose derivatives we can find by the table. Since (x100) '= 100 x99�and (sin x) '= cos x. The derivative of the sum will be equal to the sum of these derivatives:

    (x100+ sin x) '= 100 x99+ 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 = x100�- cos x

    This function is the difference of two functions whose derivatives we can also find by the table.Then the derivative of the difference is equal to the difference of the derivatives and we will not forget to change the sign, since (cos x) '= - sin x.

    (x100�- cos x) '= 100 x99�+ sin x

    Example 8. Find the derivative of the function y = ex+ tg x� x2.

    In this function there is both a sum and a difference, we find the derivatives of each term:

    (ex) '= ex, (tg x) '= 1 / cos2x, (x2) '= 2 x. Then the derivative of the original function is:

    (ex+ tg x� x2) '= ex+ 1 / cos2x �2 x

    4. Derivative work

    (f * g) '= f' * g + f * g '

    Example 9. Find the derivative of the function y = cos x * ex

    To do this, we first find the derivative of each factor (cos x) '= - sin x and (ex) '= ex. Now we will substitute everything into the formula of the work. We multiply the derivative of the first function by the second and add the product of the first function to the derivative of the second.

    (cos x * ex) '= excos x - ex* sin x

    5. Derivative private

    Then:

    (f / g) '= f' * g - f * g '/ g2

    Example 10. Find the derivative of the function y = x50/ sin x

    To find the derivative of a quotient, we first find the derivative of the numerator and denominator separately: (x50) '= 50 x49�and (sin x) '= cos x. Substituting in the derivative formula, we obtain:

    (x50/ sin x) '= 50x49* sin x - x50* cos x / sin2x

    Derivative of a complex function

    A complex function is a function represented by the composition of several functions. To find the derivative of a complex function, there is also a rule:

    (u (v)) '= u' (v) * v '

    Let's see how to find the derivative of such a function.Let y = u (v (x)) be a complex function. The function u is called external, and v is internal.

    For example:

    y = sin (x3) - complex function.

    Then y = sin (t) is an external function.

    t = x3�- internal.

    Let's try to calculate the derivative of this function. According to the formula, it is necessary to multiply the derivatives of the internal and external functions.

    (sin t) '= cos (t) is the derivative of the external function (where t = x3)

    (x3) '= 3x2- derivative of the internal function

    Then (sin (x3)) '= cos (x3) * 3x2- derivative of a complex function.


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