Rules of differentiation

The process of finding derivatives using the definition is a tedious task. We will now show methods of finding derivatives in a quick and efficient way using some simple rules.

Theorem

If  a constant, then .

Theorem

If , where n is a rational number, then

Example

Find the first derivative for

Solution

Theorem

If  have derivatives and , then

Example

Find , if

Solution

Theorem

If  and  are any two functions which have a derivative, and if then, 

Corollary

If  where c is a constant, then,

Example

If we have . Find .

Solution

Theorem

If  are any two functions which have derivative and

,

then,

Example

Find  when:

Solution

The Chain Rule

Suppose we have two functions f and g, such that g is a function of x and f is a function of g(x). The derivative of the composite function  with respect to x is obtained by the relation:

. Or,

Corollary

If , n is an integer, then,

Example

Find , given that.

Solution

Example

Let

Find .

Solution

Theorem

Corollary

Example

Find the derivative of f(x) where

Solution