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.


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.

See also

  • Derivative for parametric form at
  • 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.
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