Find d y d x of the function xy = e(x – y) - Vidyadip

Question

Find $\frac{d y}{d x}$ of the function xy = e^{(x – y)}

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Answer

Given: xy = e^{(x – y)}
Taking log on both sides, we get
log (x y) = log (e^{(x – y)})
$\Rightarrow$ log x + log y = (x - y) log e
$\Rightarrow$ log x + log y = (x - y) .1
$\Rightarrow$ log x + log y = (x - y)
Now, differentiate both sides with respect to x
$\frac{\mathrm{d}}{\mathrm{dx}} \log \mathrm{x}+\frac{\mathrm{d}}{\mathrm{dx}} \log \mathrm{y}=\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}-\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{y}$
$\implies$ $\frac{1}{x}+\frac{1}{y} \frac{d y}{d x}=1-\frac{d y}{d x}$
$\implies$$\left(1+\frac{1}{y}\right) \frac{d y}{d x}=1-\frac{1}{x}$
$\implies$$\frac{1+y}{y} \frac{d y}{d x}=\frac{x-1}{x}$
$\implies$$\frac{d y}{d x}=\frac{y(x-1)}{x(1+y)}$

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