**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.

x Constant e logx sin x cos x tan x sec x cot x cosec x sin cos tan cosec sec cot |
nx Zero e 1/x cos x -sin x sec sec x tan x -cosec -cosec x cot x 1/√(1-x -1/√(1-x 1/(1+x -1/x√(x 1/x√(x -1/(1+x |

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

**Example 1 :**

Find the derivative of x^{5} + 4x^{4} + 7x^{3} + 6x^{2} + 2 with respect to x.

**Solution :**

y = x^{5} + 4x^{4} + 7x^{3} + 6x^{2} + 2

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

= 5x^{4}+ 4x^{3} + 7x^{2} + 6x^{1}

dy/dx = 5x^{4}+ 4x^{3} + 7x^{2} + 6x^{1}

**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 }= a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

[x + (1/x)]^{3 }= x^{3} + 3x^{2}(1/x) + 3x(1/x)^{2} + (1/x)^{3}

y = x^{3} + 3x + 3/x + (1/x^{3})

y = x^{3} + 3x + 3x^{-1} + x^{-3}

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

dy/dx = 3x^{2} + 3 - 3x^{-2} -3x^{-4}

dy/dx = 3x^{2} + 3 - 3/x^{2} -3/x^{4}

**Example 4 :**

If f(x) = x^{3} − 8x + 10, find f′(x) and hence find f′(2) and f′(10).

**Solution :**

f(x) = x^{3} − 8x + 10

f'(x) = 3x^{(3-1)} - 8 (1) + 0

f'(x) = 3x^{2} - 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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