Question

If y = f (x) be a differentiable function ∀ x ∈ R, then which one of the following is always true:

• A. $\frac{d^{2}y}{dx^{2}} - \left( \frac{dx}{dy} \right)^{3}$ = 0
• B. $\frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{3}\frac{d^{2}x}{dy^{2}}$ = 0
• C. $\frac{d^{2}y}{dx^{2}} - \left( \frac{dy}{dx} \right)^{3}$ = 0
• D. None of these

Answer 1

Answer: (B)

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

Answer 2

$\frac{dy}{dx} = \left( \frac{dx}{dy} \right)^{- 1}$

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

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