If x=e^ x y then d y d x =........ - Vidyadip

MCQ

If $x=e^{\frac{x}{y}}$ then $\frac{d y}{d x}=........$

A
$1-\frac{y}{x}$
B
$1+\frac{y}{x}$
✓
$\frac{x-y}{x \log x}$
D
$\frac{x+y}{x \log x}$

✓

Answer

Correct option: C.

$\frac{x-y}{x \log x}$

$x=e^{\frac{x}{y}}$
Taking $\log$ on both sides, we get
$ \log x=\frac{x}{y}\ldots\ldots(1)$
$ \therefore y=\frac{x}{\log x}$
$\therefore \frac{d y}{d x}=\frac{\log x \times 1-x \times \frac{1}{x}}{(\log x)^2} \text {..... }[$ If $y=\frac{u}{v}$ then $\frac{d y}{d x}=\frac{v \frac{d u}{d x}-u \frac{d v}{d x}}{v^2}]$
$\therefore \frac{d y}{d x}=\frac{\frac{x}{y}-1}{\left(\frac{x}{y}\right)^2} \ldots\ldots[$ From $(1)] $
$\therefore \frac{d y}{d x}=\frac{\frac{x-y}{y}}{\frac{x \cdot x}{y} \cdot \frac{x}{y}}=\frac{x-y}{x \cdot \frac{x}{y}}$
$\therefore \frac{d y}{d x}=\frac{x-y}{x \cdot \log x} \ldots\ldots[$ From $(1)]$

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