dCdt

dC/dt, often written as dCdt, is a notation used to express the time rate of change of a quantity C with respect to time t. In scientific writing, C typically represents concentration (such as mol/L) in contexts like chemistry, pharmacokinetics, biology, and environmental science, but the concept applies to any quantity that varies over time. The derivative dC/dt captures how quickly C increases or decreases at a given moment and is fundamental to formulating dynamic models.

In mathematical terms, dC/dt is defined as the limit of the average rate of change of C over an infinitesimally small time interval. When C depends on time, dC/dt is part of a differential equation of the form dC/dt = f(C,t, …), where f describes the processes producing or removing C, such as chemical reactions, transport, production, decay, or externally imposed inputs. If the rate is proportional to the current concentration, a common model is first-order kinetics: dC/dt = -kC, yielding C(t) = C0 e^{-kt} for a constant rate k.

Applications of dC/dt include tracking drug concentration in a patient over time, modeling pollutant concentrations in air or water, and describing biochemical or physiological processes. More complex models may include terms for inflow, outflow, metabolism, binding, or compartmental transfer, leading to systems of differential equations.

Notes: while dC/dt is the standard formal notation, many texts and software environments allow the inline shorthand dCdt. Units of dC/dt are concentration per unit time, such as mol/(L·s). Understanding dC/dt is essential for analyzing dynamic behavior and solving initial-value problems in applied sciences.