dA1dt

dA1/dt is the conventional notation for the time derivative of a variable called A1. If A1 is a function of time, A1(t), then dA1/dt represents its instantaneous rate of change at time t. This quantity is a central element of differential equations that describe how systems evolve over time. In models, the evolution of A1 is typically specified by a relation such as dA1/dt = f1(A1, A2, ..., tn, t), where f1 encodes the rules governing the dynamics. A1 may represent a physical quantity such as concentration, population, amplitude, or activation level, depending on the application.

Common examples include dA1/dt = -k A1 for exponential decay with rate constant k, or dA1/dt = r A1(1 - A1/K) for logistic growth with carrying capacity K. These equations determine A1's future values given an initial condition A1(0).

In vector notation, if A is a vector of state variables, A = [A1, A2, ...], then dA/dt denotes the derivative of the vector, and its components are dA/dt = [dA1/dt, dA2/dt, ...].

Units follow the relation: the derivative has units of A1 per unit time, reflecting how quickly A1 changes. The concept is widely used across disciplines, including physics, chemistry, biology, economics, and engineering, wherever dynamic change over time is modeled.