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,

$\frac {dx} {dt}$

two very common shorthand notations are also used: adding a dot over the variable, $\dot{x}$ , 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

$\frac {d^2x} {dt^2}$

with the corresponding shorthands of $\ddot{x}$ and $x''$ .

Time derivatives are a key concept in physics. For example, for a changing position $x$ , its time derivative $\dot{x}$ is its velocity, and its second derivative with respect to time, $\ddot{x}$ , 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: