Please help me with Inverse Functions!?

Question

Suppose f-1 (f inverse) is the inverse function of a differentiable function f and f(4)=5, f'(4)=2/3. Find (f-1)'(5).

Answer 1

Using the chain rule of differentiation, it can be proved that,

[f-1]' (a) = 1/f'[f-1(a)]

Using the above formula here,

[f-1]'(5) = 1/f'[f-1(5)]

Since, f(4) = 5

that means, f-1(5) = 4

Therefore,

[f-1]'(5) = 1/f'[f-1(5)]

             = 1/f'[4]

             = 1/(2/3)           [Since f'(4) = 2/3 from the question]

             = 3/2