Derivative of an Inverse Function
How to find the derivative of an inverse function: 2 examples and their solutions.
Examplex = y5 - y + 8, dy/dx = ?
Exampley = x3 + 2, dx/dy at y = 3?
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)