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

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