JEE Main 2014 (Online) 9th AprilQuestion 1
If $y=e^{nx}$, then $\frac{d^{2}y}{dx^{2}}\cdot\frac{d^{2}x}{dy^{2}}$ is equal to :
Differentiation and its Application
Solution
Concept. $\frac{d^2x}{dy^2}$ is not $\frac{1}{d^2y/dx^2}$; compute it from $x$ as a function of $y$. Formulas used. (1) $y=e^{nx}\iff x=\dfrac{\ln y}{n}$Step 1 - $\frac{d^2y}{dx^2}$. $n^2e^{nx}$. Step 2 - $\frac{d^2x}{dy^2}$. By (1), $\frac{dx}{dy}=\frac{1}{ny}$ and $\frac{d^2x}{dy^2}=-\frac{1}{ny^2}=-\frac1n e^{-2nx}$. Step 3 - Product. $n^2e^{nx}\cdot\left(-\frac1ne^{-2nx}\right)=-ne^{-nx}$. Answer: (B)