
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 / sqrt(1  sin^{ 2}(arcsin x))
= 1 / sqrt (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 / sqrt(1  x^{ 2})
3  h^{ 1} ' (x) = 1 / (1 + x^{ 2})
More on differentiation and derivatives

