We have seen curves defined using functions, such as y = f (x). We can define more complex curves that represent relationships between x and y that are not definable by a function using parametric equations. Parametric curves are defined using two separate functions, x(t) and y(t), each representing its respective coordinate and depending on a new parameter, t. As t varies, so do the x and y coordinates of points on the curve. A curve such as y = x² can be represented parametrically by x(t) = t and y(t) = t². More complex curves involve more complex functions for x(t).
To find the rate of change of y with respect to x for a parametric curve (i.e., the first derivative with respect to x), and to find the derivative of this (i.e., the second derivative), use the following formulas:
Note that both of these derivatives are defined in terms of the parameter, t