Question:
For any differentiable function \( y \) of \( x \),
\( \dfrac{d^2 x}{dy^2} \left( \dfrac{dy}{dx} \right)^3 + \dfrac{d^2 y}{dx^2} = \)
For any differentiable function \( y \) of \( x \),
\( \dfrac{d^2 x}{dy^2} \left( \dfrac{dy}{dx} \right)^3 + \dfrac{d^2 y}{dx^2} = \)
\( \dfrac{d^2 x}{dy^2} \left( \dfrac{dy}{dx} \right)^3 + \dfrac{d^2 y}{dx^2} = \)
Show Hint
Second derivatives of inverse functions satisfy a neat cancellation identity.
Updated On: Mar 23, 2026
- \(0\)
- \(y\)
- \(-y\)
- x
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The Correct Option is A
Solution and Explanation
Step 1: Since \( \dfrac{dx}{dy} = \dfrac{1}{\dfrac{dy}{dx}} \), differentiate w.r.t. \( y \).
Step 2: Using chain rule:
\( \dfrac{d^2 x}{dy^2} = -\dfrac{d^2 y}{dx^2} \left( \dfrac{dx}{dy} \right)^3 \)
Step 3: Multiply both sides by \( \left( \dfrac{dy}{dx} \right)^3 \) to get:
\( \dfrac{d^2 x}{dy^2} \left( \dfrac{dy}{dx} \right)^3 + \dfrac{d^2 y}{dx^2} = 0 \)
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