Question

$\frac{d}{dx} (x^x)$ is equal to

• A. $x^x \, \log(e/x)$
• B. $x^x \, \log \, ex$
• C. $\log \, ex$
• D. $x^x \, \log \, x$

Answer 1

Answer: (B)

$x^x \, \log \, ex$

Answer 2

Let $y=x^{x}$
$\Rightarrow \log y=x \log x$
On differentiating w.r.t. $x$, we get
$\frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\log x$
$\Rightarrow \frac{d y}{d x}=y(1+\log x)$
$=x^{x}(\log e+\log x)$
$=x^{x}(\log e x)$