Parametric derivative

First derivative

Let x ( t ) and y ( t ) 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 x ˙ ( t ) 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:

d y d t = d y d x ⋅ d x d t

and dividing both sides by d x d t gets the equation above.

Second derivative

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

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 x ˙ and y ˙ are understood to be functions of t.