Rules of Differentiation for Algebraic Function

1 - Derivative of a constant function.

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

f '(x) = 0

Example :

f(x) = 5 , then f '(x) = 0

2 - Derivative of a power function.

The derivative of f(x) = x^n where n is a constant real number.

f '(x) = n x ^(n- 1)

Example :

f(x) = x^7

then,

f '(x) = 7 x^(7-1)

= 7x^6

3 - Derivative of the sum of functions

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

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

Example:

f(x) = 3x^4 + 2x

let g(x) = 3x^4 and h(x) = 2x

then,

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

= 12x^3 + 2

4 - 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) = 5x - x^-2

let g(x) = 5x and h(x) = x^-2

then,

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

= 5 -(-2x^-3)
= 5 + 2^-3

5 - Derivatives of a composite functions

Example :

f(x) = (2x^3 + 5)^4

let a = 2, k = 4 and n = 3

thus,

f'(x) = kanx^(n-1)(ax^n + b)^(k-1)

= 4(2)(3)x^2(2x^3 + 5)^3

= 24x^2 (2x^3 + 5)^3

6 - Derivative of the product of two functions

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

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

Example:

7 - Derivative of the quotient of two functions

Example :