If \(e^y(x + 1) = 1\) and \(\displaystyle\frac{d^2y}{dx^2}= k\left(\frac{dy}{dx}\right)^2\), then \(k\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1
Taking logarithms of \(e^y = \frac{1}{x+1}\):
\( y = -\log(x + 1) \)
Differentiating twice:
\( \displaystyle \frac{dy}{dx} = -\frac{1}{x+1} \)
\( \displaystyle \frac{d^2y}{dx^2} = \frac{1}{(x+1)^2} \)
\( \displaystyle \frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2 \)
So \(k = 1\).
NCERT (2026-27 edition):
Second Order Derivative
