Question

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

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

Answer 1

Answer: (A)

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