If \(x = y\log_e y\), where \(y \gt 0\) and \(1 + \log_e y \ne 0\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{1 + \log_e y}\)
Differentiating both sides with respect to x (implicit differentiation, product rule on the right), with y′ for \(\frac{dy}{dx}\):
\( \displaystyle 1 = y'\log_e y + y \cdot \frac{1}{y}\,y' \)
\(1 = y'(\log_e y + 1)\)
\( \displaystyle y' = \frac{1}{1 + \log_e y} \)
Option (2), \(1 + \log_e y\), is \(\frac{dx}{dy}\), the reciprocal of the required derivative.