Derivative of an Inverse Function

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

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).

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)