positionder

Positionder is a descriptive term used to refer to the derivative of a position function with respect to a parameter, most commonly time. In physics and mathematics, the derivative of position with respect to time yields the velocity, the fundamental rate of change of location.

For a position function r(t) in Euclidean space, the position derivative is the vector v(t) = dr/dt, often written as ṙ(t). If the position is parameterized by arc length s, the derivative dr/ds yields the tangent vector along the curve; when s is the physical time, this aligns with velocity (up to units). The second derivative, d^2r/dt^2, is the acceleration a(t).

In one dimension, with x(t) representing position along a line, the position derivative is v(t) = dx/dt, and the speed is |v(t)|. In higher dimensions, r(t) is a vector, and its derivative is played by velocity, with magnitude equal to the instantaneous speed.

Context and notation: r(t) ∈ R^n, v(t) = dr/dt ∈ R^n, a(t) = dv/dt ∈ R^n. Common notations include dot notation v = ṙ and acceleration a = r̈.

Applications: in physics and engineering, the position derivative is central to kinematics and dynamics; in robotics, it describes how pose changes over time; in computer graphics and animation, derivatives inform motion interpolation and easing. In any case, the existence of the derivative requires differentiability of the position function with respect to the chosen parameter.

See also: velocity, acceleration, kinematics, calculus.