Let $f$ be the function defined by $f(x)=2x+e^x$. If $g(x)=f^{-1}x$ for all $x$ and the point $(0,1)$ is on the graph of $f$, what is the value of $g'(1)$ ?

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Check the previous question's explanation for how we can derive the relationship between inverse functions.

$$g'(x)=\frac{1}{f'(g(x))}$$

Since $(0,1)$ exists on the graph of $f$, $(1,0)$ must exist on $g$.

$$f(x)=2x+e^x$$ $$f'(x)=2+e^x$$

Putting it all together:

$$g'(1)=\frac{1}{f'(g(1))}$$ $$g'(1)=\frac{1}{f'(0)}$$ $$g'(1)=\boxed{\frac{1}{3}}$$