A question on functional equations.
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} $
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?
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