\( d 2x dy 2 \) equals

\(dx/dy = 1/y'\). Then \(d 2x/dy 2 = d/dx(1/y')dx/dy = -y''/(y') 2 · 1/y' = -y''/(y') 3\) So \(d 2x/dy 2 = -(d 2y/dx 2)(dy/dx) -3 \). Answer: (D)

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Mathematics Differentiation Hard

\(\dfrac{d^2x}{dy^2}\) equals

(A) \(\left(\dfrac{d^2y}{dx^2}\right)^{-1}\)

(B) \(-\left(\dfrac{d^2y}{dx^2}\right)^{-1} \left(\dfrac{dy}{dx}\right)^{-3}\)

(C) \(\left(\dfrac{d^2y}{dx^2}\right) \left(\dfrac{dy}{dx}\right)^{-2}\)

(D) \(-\left(\dfrac{d^2y}{dx^2}\right) \left(\dfrac{dy}{dx}\right)^{-3}\)

✓ Correct Answer: Option D Verified Solution

\(\frac{dx}{dy} = \frac{1}{y'}\). Then

\(\frac{d^2x}{dy^2} = \frac{d}{dx}\left(\frac{1}{y'}\right)\frac{dx}{dy} = -\frac{y''}{(y')^2} \cdot \frac{1}{y'} = -\frac{y''}{(y')^3}\)

So \(\frac{d^2x}{dy^2} = -\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-3}\).

Answer: (D)

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