If \(xy = e^{(x-y)}\), then \(\displaystyle\frac{dy}{dx}\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{e^{x-y}-y}{x+e^{x-y}}\)
Differentiating both sides with respect to x, using the product rule on the left and the chain rule on the right:
\( \displaystyle y + x\frac{dy}{dx} = e^{x-y}\left(1 - \frac{dy}{dx}\right) \)
Collecting the \(\frac{dy}{dx}\) terms on the left:
\( \displaystyle x\frac{dy}{dx} + e^{x-y}\frac{dy}{dx} = e^{x-y} - y \)
\( \displaystyle \left(x + e^{x-y}\right)\frac{dy}{dx} = e^{x-y} - y \)
\( \displaystyle \frac{dy}{dx} = \frac{e^{x-y} - y}{x + e^{x-y}} \)
NCERT (2026-27 edition):
Logarithmic Differentiation
