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:

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