# 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