Inverse Function Rule in Differentiation

Inverse Function Rule in Differentiation

How to solve the inverse function rule in differentiation problems: formula, examples, and their solutions.

Formula

dy/dx = 1 / (dx/dy)

The reciprocal of dy/dx is dx/dy.

So it's obvious that dy/dx = 1 / (dx/dy).

So, to find dy/dx:

First find dx/dy.

Then find dy/dx
by getting the reciprocal of dx/dy.

Example 1

For the given equation, find dy/dx. x = y^3 - y + 4

Differentiate the given equation
with respect to y.

Then dx/dy = 3y2 - 1.

Derivative of polynomials

So dy/dx = 1 / (3y2 - 1).

Example 2

For the given equation, find dx/dy at y = 3. y = x^3 + 2

Differentiate the given equation
with respect to x.

Then dy/dx = 3y2.

Derivative of polynomials

So dx/dy = 1 / 3x2.

It says find dx/dy at y = 3.

But dx/dy = 1 / 3x2 has x, not y.

So find the x value when y = 3
by putting y = 3 into the given equation: y = x3 + 2.

Then x = 1.

Then [dx/dy]x = 1 = 1/3.