Question

For any differentiable function y = f(x), the value of $\frac{d^{2}y}{dx^{2}}$ +

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

= –$\left( \frac{dx}{dy} \right)^{- 2}$$\left( \frac{dy}{dx} \right)$

= –$\left( \frac{dy}{dx} \right)^{3}$ or

$\frac{d^{2}y}{dx^{2}}$ + $\frac{d^{2}x}{dy^{2}}$ = 0