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 \(x'(t)≠0\). Also, assume that \(y\) can be defined in terms of \(x\), so \(y = f(x)\) or \(y(t) = f(x(t))\). This means:
\[\begin{gather} f'(x) =\frac{dy}{dx} \\ y'(t) = f'(x(t)) x'(t) \implies f'(x(t)) = \frac{y'(t)}{x'(t)}\end{gather} \]
Since \(f'(x) = f'(x(t))\):
\[ \frac{dy}{dx} = \frac{y'(t)}{x'(t)} \]