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
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
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
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
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
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
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
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
