Question

Differentiate the following function:
$x^{\log x}+{\left(\log x\right)}^x$

Answer 1

$y=x^{\log x}+(\log x{)}^x$

$u=x^{\log x}\Rightarrow \log u+=(\log x{)}^2$

$\Rightarrow \frac{1}{u}\frac{du}{dx}=\frac{2\log \left(x\right)}{x}$

$\Rightarrow \frac{du}{dx}=\frac{2\left(\log x\right)x}{x}$

$v=(\log x{)}^x\Rightarrow \log v=x\log (\log x)$

$\Rightarrow \frac{1}{v}\frac{dv}{dx}=\frac{x}{x\log x}+\log \left(\log x\right)$

$\Rightarrow \frac{dy}{dx}=\left(\log x{)}^x(\frac{1}{\log x}+\log (\log x)\right)$

$\therefore \frac{dy}{dx}=\frac{dv}{dx}+\frac{du}{dx}$

$=\frac{2(\log x{)}^{\log x}}{x}+(\log x{)}^x(\frac{1}{\log x}+\log (\log x\left)\right)$