In calculus, a parametric derivative is a derivative of a dependent variable y with respect to an independent variable x that is taken when both variables depend on an independent third variable t, usually thought of as "time" (that is, when x and y are given by parametric equations in t ).
The first derivative implied by these parametric equations is
where the notation denotes the derivative of x with respect to t, for example. This can be derived using the chain rule for derivatives:
and dividing both sides by to give the equation above.
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:
Differentiating both functions with respect to t leads to
respectively. Substituting these into the formula for the parametric derivative, we obtain
where and are understood to be functions of t.