solve $\big(f^{-1}\big)'(0)\qquad \mbox{with}\qquad f(x)=\int_1^x\cos^2(e^t)\,dt$

Question

How to solve $$\big(f^{-1}\big)'(0)\qquad \mbox{with}\qquad f(x)=\int_1^x\cos^2(e^t)\,dt$$

I tired applying the Fundamental Theorem of Calculus but I couldn't solve it.

Answer 1

Use the formula that $(f^{-1})'(0)=\dfrac{1}{f'(f^{-1}(0))}=\dfrac{1}{f'(1)}$ since $f(1)=\displaystyle\int_{1}^{1}\cos^{2}(e^{t})dt=0$.

$f'(x)=\cos^{2}(e^{x})$, so $f'(1)=\cos^{2}(e)$.