Calculus with Parametric Curves
Calculating derivatives of parametric equations is just as easy as it is for rectangular equations.
Derivatives of Parametric Equations
When a pair of parametric equations represents the position of an object moving along the curve, $x'(t)$ and $y'(t)$ represent the horizontal velocity and vertical velocity, respectively.
The slope of the curve, then, is the ratio of vertical to horizontal velocity, $\frac{x'(t)}{y'(t)}$.
First Derivative
$\text{Let } x=f(t), y=g(t).$
$\text{If both are differentiable functions on an interval } [a,b] \text{, then: }$
$$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)}$$
$\text{provided that } \frac{dx}{dt} \ne 0.$
Second Derivative
$\text{Let } x=f(t), y=g(t).$
$$\text{If the equations } x \text{ and } y \text{ define } y \text{ as a twice-differentiable function of } x \text{,} \\
\text{then at any point where } \frac{dx}{dt} \ne 0 \text{ and } y'=\frac{dy}{dx}: \\$$
$$\frac{d^2y}{dx^2} = \frac{ \frac{d}{dt} \left( \frac{dy}{dx} \right) }{ \frac{dx}{dt} }$$