Word problem about finding the inverse derivative

Question

I have the following word problem. I need to find and interpret the meaning of the inverse derivative of a function.

At a gas station, the function f(p) is the number of gallons of gasoline sold when the price is p dollars per gallon. The manager at the station measures and finds out that

f(2)=4023 and f'(4023).

I want to find f^-1(4023) and (f^-1)'(4023). Should I just use the general formula for inverse functions? I want to see the steps on how to solve the problem.

Answer 1

Hint:

Let $y=f^{-1}(x)$ so $f(y)=x$. Now, using implicit differentiation:

$$\displaystyle\frac{d}{dx}f(y)=\frac{d}{dx}(x) \Longrightarrow f'(y)\cdot y'=1 \Longrightarrow y'=\frac{1}{f'(y)}\Longrightarrow (f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}.$$