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
An Honest Attempt
