Derivative David wants to find the rule for the derivative of f

^{-1}(x), the inverse of f(x). He knows the following derivative rules.

1. The Sum Rule: the derivative of (f+g)(x)=f(x)+g(x)

(f+g)'(x)=f'(x)+g'(x)

2. The Product Rule: the derivative of fg(x)=f(x)g(x)

(fg)'(x)=f'(x)g(x)+f(x)g'(x)

3. The Chain Rule: the derivative of (f*g)(x)=f(g(x))

(f*g)'(x)=f'(g(x))g'(x)

How can David find the Inverse Rule? What is the derivative of f^{-1}(x)?

Let f^(-1)(x) = g(x)

Then, by definition of inverse functions, we must have:

f(g(x)) = x

Differentiating both sides w.r.t x, we have:

f'(g(x))*g'(x) = 1..... by Chain Rule

Or, g'(x) = 1/f'(g(x)

Or, d/dx (f^-1(x)) = 1/f'(f^-1(x)) ....(#)

The rhs of (#) therefore, does seem to be the sought for derivative. I have been unable to simplify this further.

*Edited on ***November 27, 2021, 12:09 pm**