Differentiation of inverse function

Question

I know that first order differentiation of inverse of a function $f (x)$ is reciprocal of $f'(f^-1(x)) $. But I'm unable to evaluate the integration given in the question.

Answer 1

Supposing $e^{-x}f(x) = g(t)$ where $g(t) = 2+\int_0^1 \sqrt{1+t^4}dt$

Then $e^{-f^{-1}(x)}x = g(t)$,

Hence $-f^{-1}(x)+\ln(x) = \ln g(t)$,

$f^{-1}(x) = \ln(x)-\ln g(t)$.

Now we differentiate.

$f^{-1}{'}(x) = \frac{1}{x}$.

To check we use Inverse Function Theorem.

$f^{-1}{'}(x) = \frac{1}{f'{(f^{-1}(x))}}$.

$-e^{-x}f(x)+e^{-x}f'(x) = 0 \iff f'(x) = f(x)$.

Then $f'(f^{-1}(x)) = x$, so our inverse derivative is correct.