FIND THE DERIVATIVE OF POLYNOMIAL

Find the derivative of polynomial :

Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x.

Let us see some basic formulas used in derivatives.

Let us look into some example problems to find the derivative of the polynomial.

Example 1 :

Find the derivative of x⁵ + 4x⁴ + 7x³ + 6x² + 2 with respect to x.

Solution :

y  =  x⁵ + 4x⁴ + 7x³ + 6x² + 2

dy/dx  =  5x^(5-1)+ 4x^(4-1) + 7x^(3-1) + 6x^(2-1) + 0

  =  5x⁴+ 4x³ + 7x² + 6x¹ 

dy/dx  =  5x⁴+ 4x³ + 7x² + 6x¹ 

Example 2 :

Find the derivative of 3 sinx + 4 cosx - e^x with respect to x.

Solution :

y  =  3 sinx + 4 cosx - e^x 

dy/dx  =  3 cosx + 4 (-sinx) - e^x 

dy/dx  =  3 cosx - 4 sinx - e^x 

Example 3 :

Find the derivative of [x + (1/x)]³ with respect to x.

Solution :

y  =  [x + (1/x)]³

First, let us expand the given question using the formula (a + b)³

(a + b)³ = a³ + 3a²b + 3ab² + b³

[x + (1/x)]³  =  x³ + 3x²(1/x) + 3x(1/x)² + (1/x)³

y  =  x³ + 3x + 3/x + (1/x³)

y  =  x³ + 3x + 3x^-1 + x^-3

dy/dx  =  3x^(3-1) + 3(1) + 3(-1)x^(-1-1) + (-3)x^(-3-1)

dy/dx  =  3x² + 3 - 3x^-2 -3x^-4

dy/dx  =  3x² + 3 - 3/x² -3/x⁴

Example 4 :

If f(x) = x³ − 8x + 10, find f′(x) and hence find f′(2) and f′(10).

Solution :

f(x)  =  x³ − 8x + 10

f'(x)  =  3x^(3-1) - 8 (1) + 0

f'(x)  =  3x² - 8

From this, we have to find f'(2) and f'(10). For that we have to apply 2 and 10 instead of x in f'(x) one by one.

f'(2)  =  3(2)² - 8

f'(2)  =  3(4) - 8  ==>  12 - 8  ==>  4

f'(10)  =  3(10)² - 8

f'(10)  =  3(100) - 8  ==>  300 - 8  ==>  292

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