Differentiating Inverse functions

Steps

• Find the inverse of the function
• Differentiate the function
• Plug data into following equation:-

 

Example

 Given f(x) = x⁶, find f’(x) and state the
derivative of f^-1(x).

so

 

Example

Given f(x) = 3x^-2, find f’(x) and state the
derivative of f^-1(x).

 

 

Example

Express f(x) = x² + 2x + 4  , x ≥ 4, in the form p(x + q)² + r
Find f’(x) and state the  derivative of f^-1(x).

 

Example

Express f(x) = 4x² +2x + 4  , x ≥ 4, in the form p(x + q)² + r
Find f’(x) and state the  derivative of f^-1(x).

 

Proof

 

Differentiating Inverse Trig functions

 

Sine

Given f(x) = sinx for the interval  ,
find the derivative of f^-1(x).         

 

The given interval allows the existance of a right-angled triangle with opposite side length x units and hypotenuse 1 unit.

 

 

 

 

Cosine

Given f(x) = cosx for the interval
 , find the derivative of f^-1(x).          

The given interval allows the existance of a right-angled triangle with adjacent side length x units and hypotenuse 1 unit.

Tangent

Given f(x) = tanx for the interval
 , find the derivative of f^-1(x).         

The given interval allows the existance of a right-angled triangle with opposite side length x units and adjacent side length 1 unit.

 

Recap

 

Don’t forget the chain rule!!

Example

Integrating inverse trig functions.

 

 

Example

Find the  integral 

 

Example

Find the  integral

   

Example

 Find the  integral