Question

$\frac{d^{2}x}{d{y}^2}$ equals:

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

Answer 1

The correct option is C

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

$\frac{d^{2}x}{d{y}^2}=\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dx}\left(\frac{dx}{dy}\right)\frac{dx}{dy}$
$=\frac{d}{dx}\left(\frac{1}{\frac{dy}{dx}}\right)\frac{dx}{dy}=-\frac{1}{{\left(\frac{dy}{dx}\right)}^2}.\frac{d^{2}y}{d{x}^2}.\frac{1}{\frac{dy}{dx}}=-\frac{1}{{\left(\frac{dy}{dx}\right)}^3}\frac{d^{2}y}{d{x}^2}$