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.

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]