Consider the plane curve defined by the parametric equations \(x=x(t)\) and \(y=y(t)\). Suppose that \(x'(t)\)and \(y'(t)\) exist, and assume that \(x'(t)≠0\). The first derivative would be:
\[ \frac{d}{dx} y = \frac{dy/dt}{dx/dt} \]
If we replace \(y\) with \(dy/dx\):
\[ \frac{d^2}{dx^2} y = \frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{dx/dt} \]
Or:
\[\frac{d^2}{dx^2} y = \frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{dx/dt} \]