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)

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