calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

, a parametric derivative is a

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

of a

dependent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...

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 mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. In the case of a single parameter, parametric equations are commonly used to ...

in ''t'').

First derivative

Let and be the

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

of the points of the curve expressed as functions of a variable :

y = y(t), \quad x = x(t).

The first derivative implied by these

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

s is

\frac = \frac = \frac,

where the notation

\dot(t)

denotes the derivative of with respect to . This can be derived using the chain rule for derivatives:

\frac = \frac \cdot \frac

and dividing both sides by

\frac

to give the equation above. In general all of these derivatives — , , and — are themselves functions of and so can be written more explicitly as, for example,

Second derivative

The

second derivative In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...

implied by a parametric equation is given by

&= \frac \end

by making use of the

quotient rule In calculus, the quotient rule is a method of finding the derivative of a function (mathematics), function that is the ratio of two differentiable functions. Let h(x)=\frac, where both and are differentiable and g(x)\neq 0. The quotient rule sta ...

for derivatives. The latter result is useful in the computation of

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

Example

For example, consider the set of functions where:

\begin
x(t) &= 4t^2, &
y(t) &= 3t.
\end

Differentiating both functions with respect to leads to the functions

\begin
\frac &= 8t, &
\frac &= 3.
\end

Substituting these into the formula for the parametric derivative, we obtain

\frac = \frac = \frac,

where

\dot

and

\dot

are understood to be functions of .

References

* * * {{DEFAULTSORT:Parametric Derivative Differential calculus

First derivative

Second derivative

Example

See also

References