Parametric derivative

In calculus, a parametric derivative is a derivative 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".

For example, consider the set of functions where

:$x(t)\; =\; 4t^2\; ,$

and

:$y(t)\; =\; 3t.\; ,$

The first derivative of the parametric equations above is given by

:$frac\{frac\{dy\}\{dt\{frac\{dx\}\{dt\; =\; frac\{dot\{y\}(t)\}\{dot\{x\}(t)\},$

where the notation $dot\{x\}(t)$ 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:

:$frac\{dy\}\{dx\}\; =\; frac\{dy\}\{dt\}\; cdot\; frac\{dt\}\{dx\},$

or in other words

:$frac\{dy\}\{dx\}\; =\; frac\{frac\{dy\}\{dt\{frac\{dx\}\{dt.$

More formally, by the chain rule:

$frac\{dy\}\{dt\}\; =\; frac\{dy\}\{dx\}\; cdot\; frac\{dx\}\{dt\}$

and dividing both sides by $frac\{dx\}\{dt\}$ gets the equation above.

When we differentiate both functions with respect to "t", we end up with

:$frac\{dx\}\{dt\}\; =\; 8t$

and

:$frac\{dy\}\{dt\}\; =\; 3,$

respectively. Plugging these into the formula for the parametric derivative, we obtain

:$frac\{dy\}\{dx\}\; =\; frac\{dot\{y\{dot\{x\; =\; frac\{3\}\{8t\},$

where $dot\{x\}$ and $dot\{y\}$ are understood to be functions of "t".

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.

ee also

* Derivative (generalizations)
* Parametric equation