In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t).
First derivative[edit]
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 implied by these
parametric equations is
where the notation
denotes the derivative of
x with respect to
t. 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:
Differentiating both functions with respect to
t leads to the functions
Substituting these into the formula for the parametric derivative, we obtain
where
and
are understood to be functions of
t.
See also[edit]
References[edit]