$\frac{d}{d x}\left(x^{\ln x}+x^{\log x}\right)$

Solve:$\begin{aligned} & \frac{d}{d x}\left(x^{\ln x}+x^{\log x}\right) \\ = & \frac{d}{d x}\left(x^{\ln x}\right)+\frac{d}{d x}\left(x^{\ln x}\right) \\ = & x^{\ln x}\left[\ln x \frac{d}{d x}(\ln x)+\ln x \frac{d}{d x}(\ln x)\right] \\ + & x^{\log x}\left[\log x \frac{d}{d x}(\ln x)+\ln x \frac{d}{d x}(\log x)\right] \\ = & x^{\ln x} 2 \ln x \cdot \frac{1}{x}+x^{\log x}\left[\log x \cdot \frac{1}{x}+\ln x \cdot \frac{1}{x \ln 10}\right] \\ = & \frac{2 \ln x}{x} \cdot x^{\ln x}+x^{\log x}\left[\frac{\log x}{x}+\frac{\ln x}{x \ln 10}\right]\end{aligned}$