Time derivative

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A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as [t].

A variety of notations are used to denote the time derivative. In addition to the normal notation,

[\frac ]

two very common shorthand notations are also used: adding a dot over the variable, [\dot], and adding a prime to the variable, [x']. These two shorthands are generally not mixed in the same set of equations.

Higher time derivatives are also used: the second derivative with respect to time is written as

[\frac ]

with the corresponding shorthands of [\ddot] and [x''].

Time derivatives are a key concept in physics. For example, for a changing position [x], its time derivative [\dot] is its velocity, and its second derivative with respect to time, [\ddot], is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk.

A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another:

force is the time derivative of momentum
power is the time derivative of energy
electrical current is the time derivative of electric charge

and so on.

Time derivative

See also