# 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 Examplef(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}Examplef(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)Examplef(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)Examplef(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)Examplef(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}More on differentiation and derivatives |