Parametric differentiation

A curve in the $x-y$ plane may be the graph of a function $f(x)$ or it can be described implicitly like in the example of a unit circle given by the equation $x^{2}+y^{2}=1$ . Another way to represent a curve is through parametrisation: write $x=x(t)$ and $y=y(t)$ where $t$ is a parameter for some range of $t$ values. The dot notation is common in mechanics where $t$ denotes time. For instance,

x(t)=\cos t, \quad y(t)=\sin t, \quad \text { for } 0 \leq t \leq 2 \pi,

describes the circle $x^{2}+y^{2}=1$ . This is shown in the diagram below; the arrow indicates that $(x(t), y(t))$ moves anti-clockwise for increasing $t$ .

Circle x2 y2 = 1

To compute the slope of the tangent to the circle, we use

\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}, \quad \text { or } \quad \frac{d y}{d x}=\frac{\dot{y}}{\dot{x}},

where the dot notation denotes differentiation wrt $t$ . For the unit circle therefore,

\frac{d y}{d x}=\frac{\dot{y}}{\dot{x}}=\frac{\cos t}{-\sin t}=-\cos t

The cycloid (see diagram below) is a curve that is traced by a point on a circle as it moves along a straight line. This is described by the parametric equations

x(t)=t-\sin t, \quad y(t)=1-\cos t .

Circle x2 y2 = 1

The derivative is

\frac{d y}{d x}=\frac{\dot{y}}{\dot{x}}=\frac{\sin t}{1-\cos t} .

Note that the slope is undefined if $t$ is an integer multiple of $2 \pi$ ; these points correspond to sharp cusps in the curve.

Chain rule Leibniz's formula