Q36 / 90 · JEE Main 2011 · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'C'.

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