Find derivative without the function, only results

Question

If $g(4)=5, g’(4) = \frac23$, obtain derivative of $f^{-1} (x)$ at $x = 5$.

I’m lost on how I can get this, I tried $\frac{g’(x)}{x} but$ confused to how to get a function to plug in $5$. Am I suppose to use the line eqn $y-y_0 = m(x-x_0)$?

Thanks

Answer 1

Hint: Presumably, you want to find the derivative of the inverse function ($g=f^{-1}$).

Suppose $y=f^{-1}(x)$, and you want to find $y'$ (i.e., $dy/dx$). With sufficiently nice conditions, you have $$y=f^{-1}(x)$$ $$x=f(y)$$ $$\tfrac{d}{dx}\left[x\right]=\tfrac{d}{dx}\left[f(y)\right]$$ $$1 = f'(f(y))\cdot \tfrac{d}{dx}[y]\tag{chain rule; $y$ is a function of $x$}$$ $$1=f'(x)\cdot \tfrac{dy}{dx}\tag{$f(y)$ is $x$, from above}$$ $$\tfrac{dy}{dx} = \tfrac{1}{f'(x)}\tag{divide both sides by $f'(x)$}$$