# Parametric derivative

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* ).

## 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*, 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,

## 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*.

## See also

## External links

- Derivative for parametric form at PlanetMath.org.
- Harris, John W. & Stöcker, Horst (1998). "12.2.12 Differentiation of functions in parametric representation".
*Handbook of Mathematics and Computational Science*. Springer Science & Business Media. pp. 495–497. ISBN 0387947469.