dndT

dN/dt denotes the instantaneous rate of change of a quantity N with respect to time t. It is the derivative of N(t), the value of N at time t, with respect to t. The symbol d indicates an infinitesimal change, and dt is an infinitesimal increment in time. N may represent a count (such as a population size or number of molecules), a concentration, or any other quantity that varies with time.

Common models use dN/dt = rN for exponential growth if r > 0, or dN/dt = -λN for exponential decay if λ > 0. More realistic population models use dN/dt = rN(1 − N/K) (logistic growth), where K is carrying capacity. In chemical kinetics or nuclear processes, dN/dt can represent production minus loss; for radioactive decay, dN/dt = -λN, with λ the decay constant. In general, dN/dt = f(N,t) summarizes the net rate: the difference between rates adding to N and removing from N.

Solving the equation involves integrating with an initial condition N(t0) = N0. For linear, autonomous forms, closed-form solutions exist; e.g., dN/dt = aN + b yields N(t) = (N0 + b/a) e^{at} − b/a when a ≠ 0. In nonlinear or time-dependent cases, numerical methods or qualitative analysis are used.

The derivative has units of N per unit time. If N counts objects, dN/dt has units of objects per time. ΔN/Δt provides a discrete approximation to dN/dt for observed data.