If y = f(x) be a differentiable function ∀ x ∈ R, then which one of the following is always true:
Answer & explanation
Correct answer: option 2
We have $\frac{dy}{dx}=(\frac{dx}{dy})^{-1}$
$\frac{d^2y}{dx^2}=-(\frac{dx}{dy})^{-2}\frac{d}{dx}(\frac{dx}{dy})=-(\frac{dy}{dx})^2(\frac{d^2x}{dy^2})\frac{dy}{dx}=\frac{d^2y}{dx^2}+(\frac{dy}{dx})^3\frac{d^2x}{dy^2}=0$
Hence (B) is the correct answer.