# Inverse Function Rule in Differentiation

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

## Formula

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

Differentiate the given equation

with respect to *y*.

Then *dx*/*dy* = 3*y*^{2} - 1.

Derivative of polynomials

So *dy*/*dx* = 1 / (3*y*^{2} - 1).

## Example 2

Differentiate the given equation

with respect to *x*.

Then *dy*/*dx* = 3*y*^{2}.

Derivative of polynomials

So *dx*/*dy* = 1 / 3*x*^{2}.

It says find *dx*/*dy* at *y* = 3.

But *dx*/*dy* = 1 / 3*x*^{2} has *x*, not *y*.

So find the *x* value when *y* = 3

by putting *y* = 3 into the given equation: *y* = *x*^{3} + 2.

Then *x* = 1.

Then [*dx*/*dy*]_{x = 1} = 1/3.