- Time derivative
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**time derivative**is aderivative of a function with respect totime , usually interpreted as therate of change of the value of the function. The variable denoting time is usually written as $t,$.**Notation**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, $x\text{'},$. These two shorthands are generally not mixed in the same set of equations. Other shorthands include a subscripted "t" as in $x\_t$.

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 $x",$.

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

:$vec\; V\; =\; left\; [\; v\_1,\; v\_2,\; v\_3,\; cdots\; ight]\; ,$

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

:$frac\; \{d\; vec\; V\; \}\; \{dt\}\; =\; left\; [\; frac\{\; d\; v\_1\; \}\{dt\},frac\; \{d\; v\_2\; \}\{dt\},frac\; \{d\; v\_3\; \}\{dt\},\; cdots\; ight]\; .$

**Use in physics**Time derivatives are a key concept in

physics . For example, for a changingposition $x,$, its time derivative $dot\{x\}$ is itsvelocity , and its second derivative with respect to time, $ddot\{x\}$, is itsacceleration . Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as thejerk .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 ofmomentum

* power is the time derivative ofenergy

*electrical current is the time derivative ofelectric charge and so on.A common occurrence in physics is the time derivative of a vector, such as velocity or displacement. In dealing with such a derivative, both magnitude and orientation may depend upon time.

**Example: circular motion**For example, consider a particle moving in a circular path. Its position is given by the displacement vector $vec\; r\; =\; [x,\; y]$, related to the angle, θ, and radial distance, ρ, as defined in Figure 1:

:$x\; =\; ho\; cos\; (\; heta)\; ,$    $y\; =\; ho\; sin\; (\; heta)\; .$

For purposes of this example, time dependence is introduced by setting θ = "t". The displacement (position) at any time "t" is then:

:$vec\; r\; (t)\; =\; [\; x(t),\; y(t)\; ]\; =\; ho\; [cos\; (t),\; sin\; (t)]\; .$

The second form shows the motion described by $stackrel\{\; vec\; r\; (t)\; \}\{\}$ is in a circle of radius ρ because the "magnitude" of $vec\; r\; (t)$ is given by

:$|\; vec\; r\; |\; =\; sqrt\; \{x\; (t\; )^2\; +\; y\; (t\; )^2\; \}\; =\; ho\; left\; [\; mathrm\{cos\}^2\; (t)\; +\; mathrm\{sin\}^2\; (t)\; ight]\; ^\{1/2\}\; =\; ho\; ,$

using the result ( valid for any choice of "t" ) sin

^{2}( "t" ) + cos^{2}( "t" ) = 1.With this form for the displacement, the velocity now is found. The time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a vector made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in this case, the velocity vector is:

:$vec\; v\; (t)\; =\; frac\; \{d\; vec\; r\; (t)\; \}\{dt\}\; =\; ho\; left\; [\; frac\; \{d\; mathrm\{cos\}\; (t)\}\{dt\},\; frac\; \{d\; mathrm\; \{sin\}\; (t)\}\; \{dt\}\; ight]$

::$=\; ho\; left\; [\; -\; mathrm\{sin\}\; (t),\; mathrm\; \{cos\}\; (t)\; ight]$

::$=left\; [\; -\; y\; (t),\; x(t)\; ight]\; ,$

which shows (i) that the derivative of the displacement vector is nonzero, even though the magnitude of the displacement (the radius of the path) is constant, and (ii) that the velocity is "perpendicular" to the displacement, as can be established using the

dot product ::$vec\; v\; cdot\; vec\; r\; =\; [-y,\; x]\; cdot\; [x,\; y\; ]\; =\; -yx\; +\; xy\; =\; 0\; .$

Repeating this process for the "acceleration", which is the time derivative of the velocity:

:$vec\; a\; (t)\; =\; frac\; \{d\; vec\; v\; (t)\; \}\{dt\}\; =\; [-x(t),\; -y(t)]\; =\; -\; vec\; r\; (t)\; ,$

showing that the acceleration is inward directed, exactly opposite in direction to the displacement vector, and orthogonal to the velocity vector. This inward directed acceleration can be provided by gravitational attraction, for example, as in the case of the earth and moon, and is called centripetal acceleration.

**See also***

Differential calculus

*Notation for differentiation

*Circular motion

*Centripetal force

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