Derivative of Inverse Function
Examples with detailed solutions on how to find the derivative (Differentiation) of an inverse function, in calculus, are presented.
>

Example 1: Find the derivative dy/dx of the inverse of function f defined by
f(x)= (1/2) x  1
Solution to Example 1:
We present two methods to answer the above question
Method 1:

The first method consists in finding the inverse of f and differentiate it. To find the inverse of f we first write it as an equation
y = (1/2) x  1

Solve for x.
x = 2y + 2.

Change y to x and x to y.
y = 2x + 2.

The above gives the inverse function of f. Let us find the derivative
dy / dx = 2
Method 2:

The second method starts with one of the most important properties of inverse functions.
f(f ^{ 1}(x)) = x

Let y = f ^{ 1}(x) so that.
f(y) = x.

Differentiate both sides using chain rule to the left side.
(dy/dx)(df/dy) = 1.

Solve for dy/dx
dy / dx = 1 / (df / dy)

f is defined by
f(x)= (1/2) x  1

so that df / dy = 1/2

Substitute df / dy by 1/2 in dy / dx = 1 / (df / dy) to obtain
dy / dx = 1 / (1/2) = 2

The first method can be used only if we can find explicitly the inverse function.
Example 2: Find the derivative dy / dx where y = arcsin x.
Solution to Example 2:

arcsin x is the inverse function of sin x and
sin(arcsin(x)) = x

y = arcsin x so that
sin y = x

Differentiate both sides of the above equation, with respect to x, using the chain rule on the left side.
dy/dx cos y = 1

Solve for dy/dx.
dy/dx = 1 / cos y
= 1 / cos ( arcsin x)
= 1 / √(1  sin^{ 2}(arcsin x))
= 1 / √ (1  x^{ 2})
Exercises: Find the derivative of the inverse of each function.
1) f(x) = 3x  4
2) g(x) = arccos x
3) h(x) = arctan x
solutions to the above exercises
1) f^{ 1} ' (x) = 1 / 3
2) g^{ 1} ' (x) = 1 / √(1  x^{ 2})
3) h^{ 1} ' (x) = 1 / (1 + x^{ 2})
More on differentiation and derivatives 