Read the two statements and choose the correct option.

I. If $x = f(t)$ and $y = g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ wherever $\frac{dx}{dt} \ne 0$.

II. If $x = f(t)$ and $y = g(t)$, then $\frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2}$.

Explanation

1. 01

   I is the chain rule $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$, divided through. It is true.

2. 02

   II is false. For $x = t^2$, $y = t^3$: $\frac{dy}{dx} = \frac{3t}{2}$, so $\frac{d^2y}{dx^2} = \frac{3/2}{2t} = \frac{3}{4t}$, while the ratio in II is $\frac{6t}{2} = 3t$.