$\begin{align}  & y={{x}^{x}}+{{e}^{\tan x}} \\ & \text{Let }u={{x}^{x}},\text{ }v={{e}^{\tan x}} \\ & y=u+v \\ & \frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx} \\ & u={{x}^{x}} \\ & {{\log }_{e}}u={{\log }_{e}}{{x}^{x}} \\ & {{\log }_{e}}u=x{{\log }_{e}}x \\ & \text{Differentiate wrt }x \\ & \frac{1}{u}\frac{du}{dx}=x\left( \frac{1}{x} \right)+{{\log }_{e}}x(1) \\ & \frac{1}{u}\frac{du}{dx}=1+{{\log }_{e}}x \\ & \frac{du}{dx}=(1+{{\log }_{e}}x)u=(1+{{\log }_{e}}x){{x}^{x}} \\ & v={{e}^{\tan x}},\text{  } \\ & \frac{dv}{dx}=\frac{d}{dx}(\tan x){{e}^{\tan x}}={{\sec }^{2}}x{{e}^{\tan x}} \\ & \frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}=(1+{{\log }_{e}}x){{x}^{x}}+{{\sec }^{2}}x{{e}^{\tan x}} \\\end{align}$