Question

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

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

Answer 1

The Correct Option is C

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