Suppose f is one-to-one differentiable function...

Question

Suppose f is one-to-one differentiable function and its inverse function f^-1 is also differentiable. If f(1)=4 and f^'(1)=5, find (f^-1)'(4).

Answer 1

If b = ƒ(a) then a = ƒ^–1(b).

(The above holds if ƒ is one-to-one.)

If in addition,

ƒ '(a) = m

then

ƒ^–1 '(b) = 1/a.

BUT DO NOT write ƒ^–1(a) = 1/a. That would mean the derivative of ƒ is 1/ƒ ', and that is incorrect.