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

Question

Find $\frac{d y}{d x}$ if, :
$
x y=\log (x y)
$

✓

Answer

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

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