Question

Find the derivative of $x^x+a^x+x^a+a^a$ for some fixed $a>0,x>0$

Answer 1

Let $y=x^x+a^x+x^a+a^a$

$\frac{dy}{dx}=\frac{d}{dx}\left(x^x\right)+a^{x}lna+a{x}^{a-1}+0$

Let $u=x^x$

$lnu=xlnx$

Differentiating w.r.t -x $\Rightarrow$

$\frac{1}{u}\frac{du}{dx}=\frac{x}{x}+\left(1\right)ln\left(x\right)$

$\Rightarrow \frac{du}{dx}=x^x(1+lnx)$

$\therefore \frac{dy}{dx}=x^x(1+lnx)+x^{x}lna+a{x}^{a-1}$