Parametric derivative

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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[edit]

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[edit]

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[edit]

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[edit]

External links[edit]

  • 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.