In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t).
Let x(t) and y(t) be the coordinates of the points of the curve expressed as functions of a variable t:
In general all of these derivatives — dy / dt, dx / dt, and dy / dx — are themselves functions of t and so can be written more explicitly as, for example, .
The second derivative implied by a parametric equation is given by
For example, consider the set of functions where: