The functions $f$ and $g$ are differentiable. For all $x$, $f(g(x))=x$ and $g(f(x))=x$. If $f(3)=8$ and $f'(3)=9$, what are the values of $g(8)$ and $g'(8)$ ?

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Since we are given $f(g(x))=x$ and $g(f(x))=x$, $f$ and $g$ must be inverse functions.

Since $f(3)=8$, $\boxed{g(8)=3}$.

We can use the chain rule to obtain the formula for $g'(x)$:

$$f(g(x))=x$$ $$f'(g(x))\cdot g'(x)=1$$ $$g'(x)=\frac{1}{f'(g(x))}$$ $$g'(8)=\frac{1}{f'(g(8))}$$ $$g'(8)=\frac{1}{f'(3)}$$ $$\boxed{g'(8)=\frac{1}{9}}$$