Finding other derivatives by first principles

If       f(x) = axⁿ
then   f’(x) = anx^n-1       
a is a constant ,      n is a rational number      

 

Example

   If f(x) = 3x²  , find f’ (x)

 

 

If     f(x) = (x + a )ⁿ
then f’(x) = n(x + a)^n-1       
  a is a real number ,      n is a rational number      

 

Example

If f(x) = ( x +3)²  , find f’ (x)

 

 

If      f(x) = (ax + b )ⁿ
then  f’(x) = an(ax + b)^n-1       
 a is a real number ,  n is a rational number      

Example

If f(x) = ( 5x +3)³  , find f’ (x)  

 

 

 

The Chain Rule

 

Proof

     
         

 

 

Product Rule

 

If      f(x) = g(x).h(x) 
then  f’(x) = g’(x).h(x)+ g(x).h’(x) 

 

Example

If f(x) = 2x( 5x +3)²  , find f’ (x)  

 

 

Quotient Rule

 

This is derived from the product rule

Example

 

By just using the quotient rule