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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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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