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).

First derivative

Let and be the coordinates of the points of the curve expressed as functions of a variable The first derivative implied by these parametric equations denotes the derivative of with respect to to give the equation above.

In general, all of these derivatives — , , and — are themselves functions of and so can be written more explicitly as, for example,

Second derivative

The second derivative 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 functionsDifferentiating both functions with respect to are understood to be functions of .

See also

• Generalizations of the derivative

References