Parametric derivative

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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

$\frac{d^2y}{dx^2}$	$= \frac{d}{dx}\left(\frac{dy}{dx}\right)$
	$=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot\frac{dt}{dx}$
	$= \frac{d}{dt}\left(\frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}$
	$= \frac{\dot{x}\ddot{y} - \dot{y}\ddot{x}}{\dot{x}^3}$