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.
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.
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.
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).
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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