Parametric derivative

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Parametric derivative is a derivative in calculus that is taken when both the x and y variables (traditionally independent and dependent, respectively) depend on an independent third variable t, usually thought of as "time".

First derivative[edit]

Let and be the coordinates of the points of the curve expressed as functions of a variable t. The first derivative of the parametric equations above is given by:

where the notation denotes the derivative of x with respect to t, for example. To understand why the derivative appears in this way, recall the chain rule for derivatives:

or in other words

More formally, by the chain rule:

and dividing both sides by gets the equation above.

Second derivative[edit]

The second derivative of 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[edit]

External links[edit]