Finding derivative of function after finding inverse

Question

This is my problem:

$f(x)=2x+e^x$ and $g(x)$ is $f^{-1}(x)$ and $(0,1)$ is on the graph for $f(x)$. What is $g^{'}(x)?$

My steps

First we find inverse for function $\to x=2y+e^y$

Then do implicit differentiation to get: $2\frac{dy}{dx}+e^y\frac{dy}{dx}=1$

then plug in $y=1$.

So then I thought the answer was $\frac{1}{2+e^y}$

But the textbook tells me that the answer was $\frac{1}{3}$.

Answer 1

You're exactly right but you substituted $y=1$ when it should've been $x=1$ or $y=0$ instead, since we are looking at the inverse function now.

Answer 2

From the chain rule:

$(f^{-1})’(x) = \frac{1}{f’(f^{-1}(x))}$