$$
\begin{aligned}
& \text { } \frac{d}{d x}\left(\tan ^{-1}(\cos \sqrt{x})+\sec ^{-1}\left(\mathrm{e}^x\right)\right) \\
& =\frac{(-\sin \sqrt{x})}{1+\cos ^2 \sqrt{x}} \cdot\left(\frac{1}{2 \sqrt{x}}\right)+\frac{1}{e^x \sqrt{e^{2 x}-1}} \cdot e^x
\end{aligned}
$$
but when $x=\frac{\pi^2}{4}$
$$
\begin{aligned}
& =\frac{-\sin \frac{\pi}{2}}{1+\cos ^2 \frac{\pi}{2}}\left(\frac{1}{2}\right)\left(\frac{2}{\pi}\right)+\frac{1}{\sqrt{e^{\pi 2 / 2}-1}} \\
& =-\frac{1}{\pi}+\frac{1}{\sqrt{e^{\pi^2 / 2}-1}}
\end{aligned}
$$