Parametric derivative


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".

First derivative

Let and be the coordinates of the points of the curve expressed as functions of a variable t:
The first derivative implied by these parametric equations is
where the notation denotes the derivative of x with respect to t. 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,

Second derivative

The second derivative implied by a parametric equation is given by
by making use of the quotient rule for derivatives. The latter result is useful in the computation of curvature.

Example

For example, consider the set of functions where:
and
Differentiating both functions with respect to t leads to
and
respectively. Substituting these into the formula for the parametric derivative, we obtain
where and are understood to be functions of t.