$$
f(x)=x^{\tan x}+(\tan x)^x
$$
$$
\begin{gathered}
f_1=x^{\tan x} \Rightarrow \log f_1=\tan x \cdot \log x \\
\Rightarrow \frac{d}{d x} \log f_1=\frac{d}{d x}(\tan x \cdot \log x)
\end{gathered}
$$
$\Rightarrow \frac{d f_1}{d x}=f_1\left(\frac{\tan x}{x}+\left(\sec ^2 x\right) \log x\right)$ ...(1)
$\begin{aligned} & f_2=(\tan x)^x \Rightarrow \log f_2=x \cdot \log (\tan x) \\ & \frac{d}{d x} \log f_2=\frac{d}{d x} x \cdot \log (\tan x) \\ & \Rightarrow \frac{1}{f_2} \cdot \frac{d f_2}{d x}=\frac{x \cdot 1\left(\sec ^2 x\right)}{\tan x}+\log (\tan x)\end{aligned}$
$\Rightarrow \frac{d f_2}{d x}=f_2\left(\frac{x \sec ^2 x}{\tan x}+\log \tan x\right)$ ...(2)
$\begin{aligned} & \Rightarrow f^{\prime}(x)=\frac{d f_1}{d x}+\frac{d f_2}{d x} \\ & \Rightarrow f^{\prime}(x)=x^{\tan x}\left(\frac{\tan x}{x}+\left(\sec ^2 x\right) \log x\right) \\ & +(\tan x)^x\left(\frac{x \sec ^2 x}{\tan x}+\log \tan x\right) \\ & \Rightarrow f^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi}{4}\left(\frac{1}{\frac{\pi}{4}}+(\sqrt{2})^2 \cdot \log \frac{\pi}{4}\right)+(1)^{\pi / 4}\left(\frac{(\sqrt{2})^2 \cdot \frac{\pi}{4}}{1 \cdot}+\log 1\right) \\ & =\frac{\pi}{4}\left(\frac{4}{\pi}+2 \log \frac{\pi}{4}\right)+\frac{\pi}{2}=1+\frac{\pi}{2} \log \frac{\pi}{4}+\frac{\pi}{2} \\ & =1+\frac{\pi}{2}\left(\log \frac{\pi}{4}+1\right)=1+\frac{\pi}{2}\left(\log \frac{\pi}{4}+\log e\right) \\ & =1+\frac{\pi}{2} \log \left(\frac{e \pi}{4}\right) . \\ & \end{aligned}$