Inverse Functions and proving the product of derivatives is 1

Question

Write down the derivative of the function y=x^3 - 1: 3x^2

Make x the subject and hence find dx/dy: 1/3(y+1)^-2/3

Show that dy/dx * dx/dy = 1

How does the x and y possibly cancel out?

It seems like a very straightforward question but what do i do?

Answer 1

Well, we have

$$\frac{dx}{dy}=\frac13(y+1)^{-2/3}=\frac13((x^3-1)+1)^{-2/3}=\frac13x^{-2}$$

Then can you show the product is one?