Find d y d x if, : y=x^ e^x - Vidyadip

Question

Find $\frac{d y}{d x}$ if, :
$
y=x^{e^x}
$

✓

Answer

$
\begin{aligned}
& y=x^{e^x} \\
& \therefore \log y=\log x^{e^x}=e^x \cdot \log x
\end{aligned}
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
\frac{1}{y} \cdot \frac{d y}{d x} & =\frac{d}{d x}\left(e^x \cdot \log x\right) \\
& =e^x \cdot \frac{d}{d x}(\log x)+(\log x) \cdot \frac{d}{d x}\left(e^x\right) \\
& =e^x \cdot \frac{1}{x}+(\log x)\left(e^x\right) \\
\therefore \frac{d y}{d x} & =y\left[\frac{e^x}{x}+e^x \cdot \log x\right] \\
& =x^{e^x} \cdot e^x\left[\frac{1}{x}+\log x\right]
\end{aligned}
$

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