Question

Differentiate $(\log x{)}^x+x^{\log x}$ with respect to $x$.

Answer 1

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

Let $y=a+b$

$\frac{dy}{dx}=$?

$a={\left(\log x\right)}^x$

$\Rightarrow \log a=x\log x\Rightarrow \frac{1}{a}\frac{da}{dx}=\log x+\frac{x}{x}=1+\ln x$

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

$b=x^{\log x}$

$\log b=\left(\log x\right).x$

$\frac{1}{b}.\frac{db}{dx}=\frac{1}{x}.x+\log x$

$\frac{db}{dx}=b(1+\log x)=x^{\log x}(1+\log x)$

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

$\frac{dy}{dx}=(1+\log x)y$