Derivative of a function with respect to time
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 formula_1.
Notation.
A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,
formula_2
A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E.
formula_3
Higher time derivatives are also used: the second derivative with respect to time is written as
formula_4
with the corresponding shorthand of formula_5.
As a generalization, the time derivative of a vector, say:
formula_6
is defined as the vector whose components are the derivatives of the components of the original vector. That is,
formula_7
Use in physics.
Time derivatives are a key concept in physics. For example, for a changing position formula_8, its time derivative formula_3 is its velocity, and its second derivative with respect to time, formula_5, 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.
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.
Example: circular motion.
For example, consider a particle moving in a circular path. Its position is given by the displacement vector formula_11, related to the angle, "θ", and radial distance, "r", as defined in the figure:
formula_12
For this example, we assume that "θ" = "t". Hence, the displacement (position) at any time "t" is given by
formula_13
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
formula_14
using the trigonometric identity sin2("t") + cos2("t") = 1 and where formula_15 is the usual Euclidean dot product.
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:
formula_16
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:
formula_17
Acceleration is then the time-derivative of velocity:
formula_18
The acceleration is directed inward, toward the axis of rotation. It points opposite to the position vector and perpendicular to the velocity vector. This inward-directed acceleration is called centripetal acceleration.
In differential geometry.
In differential geometry, quantities are often expressed with respect to the local covariant basis, formula_19, where "i" ranges over the number of dimensions. The components of a vector formula_20 expressed this way transform as a contravariant tensor, as shown in the expression formula_21, invoking Einstein summation convention. If we want to calculate the time derivatives of these components along a trajectory, so that we have formula_22, we can define a new operator, the invariant derivative formula_23, which will continue to return contravariant tensors:
formula_24
where formula_25 (with formula_26 being the "j"th coordinate) captures the components of the velocity in the local covariant basis, and formula_27 are the Christoffel symbols for the coordinate system. Note that explicit dependence on "t" has been repressed in the notation. We can then write:
formula_28
as well as:
formula_29
In terms of the covariant derivative, formula_30, we have:
formula_31
Use in economics.
In economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives.ch. 1-3 One situation involves a stock variable and its time derivative, a flow variable. 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:
References.
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