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

Question

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

✓

Answer

$
\begin{aligned}
& x ^{ y }= e ^{( x - y )} \\
& \therefore \log x ^{ y }=\log e ^{( x - y )} \\
& \therefore y \log x =( x - y ) \log e \\
& \therefore y \log x = x - y \ldots \ldots[\because \log e =1] \\
& \therefore y + y \log x = x \\
& \therefore y (1+\log x )= x \\
& \therefore y =\frac{x}{1+\log x} \\
& \therefore \frac{d y}{d x}=\frac{d}{d x}\left(\frac{x}{1+\log x}\right) \\
& \qquad \frac{(1+\log x) \cdot \frac{d}{d x}(x)-x \frac{d}{d x}(1+\log x)}{(1+\log x)^2}
\end{aligned}
$
$
\begin{aligned}
& =\frac{(1+\log x) \cdot 1-x\left(0+\frac{1}{x}\right)}{(1+\log x)^2} \\
& =\frac{1+\log x-1}{(1+\log x)^2} \\
& =\frac{\log x}{(1+\log x)^2} .
\end{aligned}
$

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