Question

If \( y = f(x) \) is a thrice differentiable function and a bijection, then \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = ? \]

• A. \( y \)
• B. \( -y \)
• C. \( x \)
• D. \( 0 \)

Hint:

For differentiable bijections, inverse differentiation follows: \[ \frac{dx}{dy} = \left(\frac{dy}{dx}\right)^{-1} \] and applying the second derivative relation helps in solving such problems.

Answer 1

The Correct Option is D

Step 1: Differentiating Implicitly We start with the given equation: \[ \frac{dx}{dy} = \left(\frac{dy}{dx}\right)^{-1} \] Differentiating both sides with respect to \( y \): \[ \frac{d^2x}{dy^2} = -\left(\frac{dy}{dx}\right)^{-2} \cdot \frac{d^2y}{dx^2} \] Multiplying by \( \left(\frac{dy}{dx}\right)^3 \): \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 = -\frac{d^2y}{dx^2} \] Rearranging: \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = 0 \]