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

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

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.