Need Help If f(x) = x + x^3, what is the derivative of f^-1(s) at s = 2

Question

If f(x) = x + x³, what is the derivative of f ^-1(s) at s = 2

Answer 1

What you need is the formula for the derivative of f^-1 in terms of f and f'.

Note: this is a completely edited answer...my original answer was not correct.

This method of finding the derivative of the inverse function is probably not in standard form;

it does work, however.

y = x+ x³

Interchange the roles of x and y:

x = y + y³

Differentiate implicitly

1 = dy/dx + 3y² dy/dx...where dy/dx is the derivative of the INVERSE function

and y = f^-1(x)

dy/dx = 1/(1 + 3y²)

I am not sure what is wanted, but if y=2, the slope of the tangent to the inverse function is 1/12