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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability2 Marks
Question
Find $\frac{d y}{d x}$ of the function $xy = e^{(x – y)}$
✓

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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