Question

For any differentiable function y = f(x), the value of

$\frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{3}\frac{d^{2}x}{dy^{2}}$ is

• A. Always zero
• B. Always non-zero
• C. Equal to 2y²
• D. Equal to x²

Answer 1

Answer: (A)

Always zero

Answer 2

$\left( \frac{dy}{dx} \right) = \left( \frac{dx}{dy} \right)^{–1}$ for a differentiable coefficient

or $\frac{d^{2}y}{dx^{2}} = –1\left( \frac{dx}{dy} \right)^{–2}\frac{d}{dy}\left( \frac{dx}{dy} \right)\frac{dy}{dx}$ =$–\left( \frac{dx}{dy} \right)^{–2}\frac{d^{2}x}{dy^{2}}\left( \frac{dy}{dx} \right)$

= –$\frac{d^{2}x}{dy^{2}}\left( \frac{dy}{dx} \right)^{3}$ or $\frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{3}\frac{d^{2}x}{dy^{2}} = 0$