dYdT

dydt denotes the derivative of y with respect to time t. It is typically written as dy/dt and represents the instantaneous rate of change of the dependent quantity y as time advances. In physics and engineering, the dot notation ẏ is also used to indicate the same time derivative.

Formally, if y(t) is differentiable at t, dydt is defined by the limit dy/dt = lim_{h→0} [y(t+h) − y(t)]/h. This limit measures how rapidly y changes at that moment. When y is a function of time only, dydt describes a one-dimensional rate of change; when y is a function of several variables, ∂y/∂t is used to denote the partial derivative with respect to time.

In ordinary differential equations, dydt often appears in the form dy/dt = f(t, y), where the rate of change of y depends on time and the current value of y. Solutions to such equations are trajectories describing how y evolves over time. The notation y′(t) or ẏ(t) is common in mathematical and applied contexts, and dy/dt is frequently preferred in more formal writing.

Distinguishing dy/dt from related notions is important. dy/dt refers to a derivative with respect to time, whereas ∂y/∂t denotes a partial derivative when y is a function of multiple variables. In discrete-time models, finite differences Δy/Δt approximate dydt rather than its exact continuous limit.

Applications of dydt span physics (velocity as the time derivative of position), chemistry (rates of concentration change), biology (growth rates), economics (time-based changes in variables), and engineering (dynamic system modeling).