d^2 x d y^2 equals - Vidyadip

MCQ

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

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

✓

Answer

Correct option: D.

$-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}$

d
Since, $\frac{d x}{d y}=\frac{1}{d y / d x}=\left(\frac{d y}{d x}\right)^{-1}$

$ \Rightarrow \frac{d}{d y}\left(\frac{d x}{d y}\right)=\frac{d}{d x}\left(\frac{d y}{d x}\right)^{-1} \frac{d x}{d y} $

$ \Rightarrow \frac{d^2 x}{d y^2}=-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-2}\left(\frac{d x}{d y}\right)=-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}.$

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