Calculus of a function and its inverse

Question

Suppose f(x) is an invertible, differentiable function and f(3) = -3, f(-3) = 1,

f '(3) = 4, and f '(1) = 5.

Find (f^-1 ) ' (-3), that is, find the value of the derivative of f ^-1 (x) at x = -3.

Answer 1

For notational convenience, use g(x) as the name for the inverse of f(x), i.e. 

g(x) = f ^-1 (x).

Recall that derivatives for a function and its inverse are related by:  

g ‘ (x) = ____1____

                  f ‘ [g(x)]

Since f ^-1 and g are the same function, the question translates to:  find g’(-3)

Substitute x = -3 into our inverse equation above:

g’(-3) = ____1____

                  f ‘ [g(-3)]

By the nature of inverses functions, since f(3) = -3, then g(-3) = 3

g’(-3) = ____1____  = ____1____     since f ‘ (-3) = 4 (given)

                  f ‘ (-3)                 4