Parametric Equations and Their Derivatives

A pair of functions $x(t)$ and $y(t)$ traces a curve: at each value of the parameter $t$, the point $(x(t), y(t))$ is on it. The curve has a direction, the way the point travels as $t$ increases. It can loop and cross itself in ways the graph of a function can't.

Where $y$ is a differentiable function of $x$ along the curve, the chain rule gives $\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$. Solving for $\frac{dy}{dx}$ gives the slope in terms of $t$.

The tangent line is horizontal where $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \ne 0$, and vertical where $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \ne 0$. Where both are zero, the formula says nothing, and the curve needs a closer look.

The slope $\frac{dy}{dx}$ is itself a function of $t$. Its rate of change with respect to $x$ comes from the same rule, applied to $\frac{dy}{dx}$ in place of $y$.

The derivative of $\frac{dy}{dx}$ with respect to $t$ is only the top of the fraction. Dividing by $\frac{dx}{dt}$ converts it to a rate with respect to $x$. Its sign gives the concavity of the curve.

When a curve passes through the same point at two values of $t$, it can have a different tangent line on each pass.