JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Derivatives XI

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

equals

Answer: (D)

Solution

We know that

\frac{d x}{d y} = (\frac{d y}{d x})^{- 1}

.

Differentiating both sides with respect to

y

\frac{d ^{2} x}{d y ^{2}} = \frac{d}{d y} [(\frac{d y}{d x})^{- 1}] = \frac{d}{d x} [(\frac{d y}{d x})^{- 1}] \cdot \frac{d x}{d y}

Using the chain rule:

\frac{d ^{2} x}{d y ^{2}} = - (\frac{d y}{d x})^{- 2} \cdot \frac{d ^{2} y}{d x ^{2}} \cdot (\frac{d y}{d x})^{- 1} = - (\frac{d ^{2} y}{d x ^{2}}) (\frac{d y}{d x})^{- 3}

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Single correct option