Question

$\frac{d^{2}x}{d{y}^2}$ is equal to (IIT-JEE, 2007)

• A. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}$
• B. $\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$
• C. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-2}$
• D. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

Answer 1

The correct option is D $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

$\frac{d^{2}x}{d^{2}y}=\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dx}\left(\frac{dx}{dy}\right)\frac{dx}{dy}$

$=\left\{\frac{d}{dx}\left[\frac{1}{\left(\frac{dy}{dx}\right)}\right]\right\}\frac{1}{\left(\frac{dy}{dx}\right)}=-\frac{1}{{\left(\frac{dy}{dx}\right)}^2}\frac{d^{2}y}{d{x}^2}\frac{1}{\left(\frac{dy}{dx}\right)}$

$={\left(\frac{dy}{dx}\right)}^{-3}\frac{d^{2}y}{d{x}^2}$