Derivative of an Inverse Function

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

Example 1

Example

The given function is x = f(y), not y = f(x).
So it says to find the derivative of an inverse function.

Solution

x = y5 - y + 8

So dx/dy = 5y4 - 1.

Derivative of a Polynomial

dy/dx = 1/(dx/dy)

dx/dy = 5y4 - 1

So dy/dx = 1/(5y4 - 1).

So dy/dx = 1/(5y4 - 1).

Example 2

Example

Solution

y = x3 + 2

So dy/dx = 3x2.

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

dy/dx = 3x2

So dx/dy = 1/3x2.

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

But dx/dy = 1/3x2 doesn't have y.
So you can't put y = 3 into dx/dy = 1/3x2.

So, to find the right x,
put y = 3 into the original function y = x3 + 2.

Then 3 = x3 + 2.

Then x = 1.

Put x = 1 into dx/dy = 1/3x2.

Then [dx/dy]x = 1 = 1/(3⋅12).

Then you get 1/3.

So dx/dy = 1/3 at y = 3. (at x = 1)