dVdt

dv/dt, read as “the derivative of v with respect to t,” is a fundamental concept in calculus used to express the instantaneous rate of change of a quantity v as time t changes. When v represents velocity, dv/dt is the acceleration, the rate at which velocity changes. More generally, dv/dt describes how any dependent variable v changes over time.

Notation and meaning: The derivative is defined mathematically as dv/dt = lim Δt→0 (Δv/Δt). For vector-valued v, dv/dt is the time derivative taken componentwise, yielding a vector. If v is itself a function of t through other variables, the derivative follows the standard rules of differentiation (chain rule, product rule). In many texts dv/dt is abbreviated as a, especially when v is velocity, but dv/dt remains valid for any time-dependent quantity. The concatenated form dvdt is common in informal writing or programming contexts but is not a standard mathematical symbol in formal notation.

Applications: dv/dt appears across physics and engineering, in motion analysis, control theory, and differential equations. It is used to relate velocity changes to applied forces, to model how systems evolve over time, and to formulate equations of motion. Higher-order derivatives—such as d^2v/dt^2, the derivative of acceleration—describe the rate at which acceleration changes (jerk).

Limitations: The derivative dv/dt exists only if v is differentiable with respect to time at the point of interest. If v is discontinuous or non-differentiable, dv/dt may be undefined.

See also: derivative, rate of change, velocity, acceleration, calculus, differential equations.