Time derivative

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 ${\displaystyle t\,}$.

A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,

A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E.

(This is called Newton's notation)

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

with the corresponding shorthand of ${\displaystyle {\ddot {x}}}$.

As a generalization, the time derivative of a vector, say:

is defined as the vector whose components are the derivatives of the components of the original vector. That is,

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

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