# Derivative of an Inverse Function

How to find the derivative of an inverse function: 2 examples and their solutions.

## Examplex = y^{5} - y + 8, dy/dx = ?

The given function is x = f(y), not y = f(x).

So it says to find the derivative of an inverse function.

x = y^{5} - y + 8

So dx/dy = 5y^{4} - 1.

Derivative of a Polynomial

dy/dx = 1/(dx/dy)

dx/dy = 5y^{4} - 1

So dy/dx = 1/(5y^{4} - 1).

So dy/dx = 1/(5y^{4} - 1).

## Exampley = x^{3} + 2, dx/dy at y = 3?

y = x^{3} + 2

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

dx/dy = 1/(dy/dx).

dy/dx = 3x^{2}

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

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

But dx/dy = 1/3x^{2} doesn't have y.

So you can't put y = 3 into dx/dy = 1/3x^{2}.

So, to find the right x,

put y = 3 into the original function y = x^{3} + 2.

Then 3 = x^{3} + 2.

Then x = 1.

Put x = 1 into dx/dy = 1/3x^{2}.

Then [dx/dy]_{x = 1} = 1/(3⋅1^{2}).

Then you get 1/3.

So dx/dy = 1/3

at y = 3. (at x = 1)