A question on functional equations.

Question

Question:

If it is given that $$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$

then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $

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Where I am stuck:

Now, since we are to evaluate $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $, all we need to evaluate is $ f'(2) $. Our answer will be the reciprocal of this.

So I differentiated the given equation:

$$ e^x(f(x) + f'(x)) = \sqrt{1+x^4}\ $$

But to find $f'(2)$ we also need to evaluate $f(2)$ which seems to be an insane thing to do. So what to do?

Answer 1

Hint: We don't need $f(2)$, we need $f^{-1}(2)$, which is $0$.

Now go ahead and find $f'(0)$, like you (sort of) started to do.