# Rules of Differentiation of Functions in Calculus

The basic rules of Differentiation of functions in calculus are presented along with several examples .

## 1 - Derivative of a constant function.

The derivative of f(x) = c where c is a constant is given by

**f '(x) = 0**

__Example__

f(x) = - 10 , then f '(x) = 0

## 2 - Derivative of a power function (power rule).

The derivative of f(x) = x

^{ r}where r is a constant real number is given by

**f '(x) = r x**

^{ r - 1}__Example__

f(x) = x

^{ -2}, then f '(x) = -2 x

^{ -3}= -2 / x

^{ 3}

## 3 - Derivative of a function multiplied by a constant.

The derivative of f(x) = c g(x) is given by

**f '(x) = c g '(x)**

__Example__

f(x) = 3x

^{ 3},

let c = 3 and g(x) = x

^{ 3}, then f '(x) = c g '(x)

= 3 (3x

^{ 2}) = 9 x

^{ 2}

## 4 - Derivative of the sum of functions (sum rule).

The derivative of f(x) = g(x) + h(x) is given by

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

__Example__

f(x) = x

^{ 2}+ 4

let g(x) = x

^{ 2}and h(x) = 4, then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x

## 5 - Derivative of the difference of functions.

The derivative of f(x) = g(x) - h(x) is given by

**f '(x) = g '(x) - h '(x)**

__Example__

f(x) = x

^{ 3}- x

^{ -2}

let g(x) = x

^{ 3}and h(x) = x

^{ -2}, then

f '(x) = g '(x) - h '(x) = 3 x

^{ 2}- (-2 x

^{ -3}) = 3 x

^{ 2}+ 2x

^{ -3}

## 6 - Derivative of the product of two functions (product rule).

The derivative of f(x) = g(x) h(x) is given by

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

__Example__

f(x) = (x

^{ 2}- 2x) (x - 2)

let g(x) = (x

^{ 2}- 2x) and h(x) = (x - 2), then

f '(x) = g(x) h '(x) + h(x) g '(x) = (x

^{ 2}- 2x) (1) + (x - 2) (2x - 2)

= x

^{ 2}- 2x + 2 x

^{ 2}- 6x + 4 = 3 x

^{ 2}- 8x + 4

## 7 - Derivative of the quotient of two functions (quotient rule).

The derivative of f(x) = g(x) / h(x) is given by

**f '(x) = ( h(x) g '(x) - g(x) h '(x) ) / h(x)**

^{ 2}__Example__f(x) = (x - 2) / (x + 1)

let g(x) = (x - 2) and h(x) = (x + 1), then

f '(x) = ( h(x) g '(x) - g(x) h '(x) ) / h(x)

^{ 2}

= ( (x + 1)(1) - (x - 2)(1) ) / (x + 1)

^{ 2}

= 3 / (x + 1)

^{ 2}