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.

[1] The variable denoting time is usually written as

A variety of notations are used to denote the time derivative.

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

That is, Time derivatives are a key concept in physics.

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: 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.

For example, consider a particle moving in a circular path.

, related to the angle, θ, and radial distance, r, as defined in the figure: For this example, we assume that θ = t. Hence, the displacement (position) at any time t is given by This form shows the motion described by r(t) is in a circle of radius r because the magnitude of r(t) is given by using the trigonometric identity sin2(t) + cos2(t) = 1 and where

Thus, in this case, the velocity vector is: Thus the velocity of the particle is nonzero even though the magnitude of the position (that is, the radius of the path) is constant.

The velocity is directed perpendicular to the displacement, as can be established using the dot product: Acceleration is then the time-derivative of velocity: The acceleration is directed inward, toward the axis of rotation.

In differential geometry, quantities are often expressed with respect to the local covariant basis,

If we want to calculate the time derivatives of these components along a trajectory, so that we have

being the jth coordinate) captures the components of the velocity in the local covariant basis, and

Note that explicit dependence on t has been repressed in the notation.

, we have: In economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives.[3]: ch.

Examples include: Sometimes the time derivative of a flow variable can appear in a model: And sometimes there appears a time derivative of a variable which, unlike the examples above, is not measured in units of currency: