Finding derivative of inverse function.

Question

I was trying the below question and am unable to integrate the function. I tried different substitutions but get stuck in between.

Let $f(x)=\displaystyle\int_2^x\frac{1}{1+t^4}\,dt$ and $g(x)=f^{-1}(x)$. Find $\frac{d}{dx}(g(x))$ at x=0.

Answer 1

Since $x=f\left( g\right)$, $\frac{dx}{dg}=f'\left( g\right)=\frac{1}{1+g^4}$ so $\frac{dg}{dx}=\left( \frac{dx}{dg}\right)^{-1}=1+g^4$. Solving $f\left( g\right)=0$ gives $g=2$, so at $x=0$ we have $\frac{dg}{dx}=17$.