Solve this


If $y=x^{x}$, find $\frac{d y}{d x}$ at $x=e$.



Taking logarithm on both sides,

$\log y=x \log x$

Differentiating w.r.t. $x$ on both sides,

$\frac{1}{y} \cdot \frac{d y}{d x}=x \cdot \frac{1}{x}+1 \cdot \log x$

$=1+\log x$

$\Rightarrow \frac{d y}{d x}=y(1+\log x)$

$=x^{x}(1+\log x)$

So, at $x=e$,

$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\mathrm{e}}(1+\log \mathrm{e})$


$=2 e^{e}($ Ans $)$

Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now