Time derivative

A time derivative measures how a quantity changes as time changes. It is obtained by differentiating the quantity with respect to a time variable.

Derivative: A derivative measures the instantaneous rate of change of a function with respect to its input. A time derivative is the derivative whose input is time.
Function: A rule or relation assigning each allowed input exactly one output. A time-varying quantity is represented as a function whose input is time.
Newton's second law: A law stating that an object's net force equals the rate of change of its momentum. For constant mass, it relates force to the second time derivative of position.
Antiderivative: A function whose derivative equals a specified function. It reverses differentiation and recovers quantities from a rate, up to a constant.
Difference quotient: A ratio of changes in a function and its input over a finite interval. The time derivative is the limiting value of this ratio as the time interval shrinks.
Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified point. The instantaneous rate is defined by a limit of finite-time change ratios.
Angular velocity: The rate of change of angular position with respect to time. It applies time differentiation to rotational rather than linear position.
Finite difference: A difference between function values at distinct inputs, often divided by their spacing. It approximates a time derivative but measures change across a nonzero interval.
Velocity: The rate of change of position with respect to time, including direction. Velocity is the time derivative of position.
Leibniz's notation: A notation for derivatives that writes changes in dependent and independent variables as differentials. The symbol dy/dt explicitly identifies time as the variable of differentiation.

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