Maths. Section AContinuity and Differentiability

If e^y(1+x)=1 , then (d^2y)/(dx^2) is:

If $ e^y(1+x)=1 $, then $ \displaystyle\frac{d^2y}{dx^2} $ is:

Answer & explanation

Correct answer: option 1

The correct answer is Option (1) → $ \left(\displaystyle\frac{dy}{dx}\right)^2 $

Taking logarithms of $ e^y(1+x)=1 $ gives y explicitly:

$ y+\log(1+x)=0 $

$ y=-\log(1+x) $

$ \displaystyle \frac{dy}{dx}=-\frac{1}{1+x} $

$ \displaystyle \frac{d^2y}{dx^2}=\frac{1}{(1+x)^2} $

$ \displaystyle =\left(-\frac{1}{1+x}\right)^2 $

$ \displaystyle =\left(\frac{dy}{dx}\right)^2 $

NCERT (2026-27 edition):
Second Order Derivative

$NCERT extract$