Question

Rules of differentiation with example

Answer 1

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