Question

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

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

Answer 1

Answer: (A)

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