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AIEEE 2007

JEE Main 2007 Mathematics question on Limits Continuity and Differentiability

d2xdy2\frac{d^2x}{dy^2} equals
  1. A.
    (d2ydx2)−1\left(\frac{d^2y}{dx^2}\right)^{-1}
  2. B.
    −(d2ydx2)−1(dydx)−3-\left(\frac{d^2y}{dx^2}\right)^{-1}\left(\frac{dy}{dx}\right)^{-3}
  3. C.
    (d2ydx2)(dydx)−2\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-2}
  4. D.
    −(d2ydx2)(dydx)−3-\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-3}
Show answer and solution

Correct option: D

Solution

Start with dx/dy = (dy/dx)⁻¹. Differentiate both sides with respect to y:

d²x/dy² = d/dy[(dy/dx)⁻¹] = d/dx[(dy/dx)⁻¹] · dx/dy

= -(dy/dx)⁻² · (d²y/dx²) · (dy/dx)⁻¹

= -(d²y/dx²)(dy/dx)⁻³

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