Question

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