Question: how to find the inverse of a function using derivatives?
Answer: Use the inverse function theorem to find the derivative of g(x)=3√x.
The function g(x)=3√x is the inverse of the function f(x)=x3. Since g′(x)=1f′(g(x)), begin by finding f′(x). Thus,
f′(x)=3x3.
f′(g(x))=3(3√x)2=3x2/3.
g′(x)=13x2/3=13x−2/3.