Question

For any differentiable function $y$ of $x$,
$\frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=$

• A. $0$
• B. $y$
• C. $-y$
• D. $x$

Answer 1

Answer: (A)

$0$

$\frac{dy}{dx}={\left(\frac{dx}{dy}\right)}^{-1}\Rightarrow \frac{d^{2}y}{d{x}^2}=-1{\left(\frac{dx}{dy}\right)}^{-2}\left\{\frac{d}{dx}\left(\frac{dx}{dy}\right)\right\}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=(-1){\left(\frac{dx}{dy}\right)}^{-2}\left\{\frac{d}{dy}\left(\frac{dx}{dy}\right)\frac{dy}{dx}\right\}$
$=(-1){\left(\frac{dy}{dx}\right)}^2\{\frac{d^{2}x}{d{y}^2}\cdot \frac{dy}{dx}\}=-{\left(\frac{dy}{dx}\right)}^3\left\{\frac{d^{2}x}{d{y}^2}\right\}$
$\Rightarrow \frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=0$.