Parametric derivative

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

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:

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

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.

Second derivative

The second derivative of a parametric equation is given by

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

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:

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

and

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

Differentiating both functions with respect to t leads to

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

and

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

respectively. Substituting 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.

See also

• Derivative (generalizations)
• Parametric equation

External links

• Derivative for parametric form at PlanetMath.org.
• Harris, John W. & Stöcker, Horst (1998). "12.2.12 Differentiation of functions in parametric representation". Handbook of Mathematics and Computational Science. Springer Science & Business Media. pp. 495–497. ISBN 0387947469.