# FIND THE DERIVATIVE OF POLYNOMIAL

## About "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.

 xnConstantexlogxsin xcos xtan xsec xcot xcosec xsin-1xcos-1xtan-1xcosec-1xsec-1xcot-1x nx(n-1)Zeroex1/xcos x-sin xsec2 xsec x tan x-cosec2 x-cosec x cot x1/√(1-x2)-1/√(1-x2)1/(1+x2)-1/x√(x2-1)1/x√(x2-1)-1/(1+x2)

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 - ex with respect to x.

Solution :

y  =  3 sinx + 4 cosx - ex

dy/dx  =  3 cosx + 4 (-sinx) - ex

dy/dx  =  3 cosx - 4 sinx - ex

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)= 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 After having gone through the stuff given above, we hope that the students would have understood "Find the derivative of polynomial".

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