If \(x^y=e^{x-y}\), then \(\displaystyle\frac{dy}{dx}=\)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\log x}{(1+\log x)^2}\)
Taking logarithms of both sides:
\(y\log x = x - y\)
\(y(1+\log x) = x\)
\( \displaystyle y = \frac{x}{1+\log x} \)
By the quotient rule:
\( \displaystyle \frac{dy}{dx} = \frac{(1+\log x)\cdot 1 - x\cdot\frac{1}{x}}{(1+\log x)^2} \)
\( \displaystyle = \frac{\log x}{(1+\log x)^2} \)
NCERT Exemplar:
Continuity and Differentiability
