Time derivative

The time derivative is the derivative of a function with respect to time , usually interpreted as the rate of change of the value of a function. ^[1] Time is usually indicated by a variable. $t$ ${\ displaystyle t}$ $t$ .

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Legend

To refer to the time derivative, several notations are used. In addition to the usual (leibnitz) notation,

{\frac {dx}{dt}}

{\ displaystyle {\ frac {dx} {dt}}}

Very often, especially in physics, an abbreviated notation with a dot above the variable is used:

{\dot {x}}

{\ displaystyle {\ dot {x}}}

(so-called Newtonian notation).

Higher time derivatives are denoted as:

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

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

or in abbreviated form: ${\ddot {x}}$ ${\ displaystyle {\ ddot {x}}}$ .

In the case of time derivatives of higher orders, the Newtonian notation, as a rule, is not used.

More generally, the time derivative of a vector is:

{\vec {V}}=\left[v_{1},\ v_{2},\ v_{3},\cdots \right]\ ,

{\ displaystyle {\ vec {V}} = \ left [v_ {1}, \ v_ {2}, \ v_ {3}, \ cdots \ right] \,}

is defined as a vector with components that are derived from the corresponding components of the original vector. I.e

{\frac {d{\vec {V}}}{dt}}=\left[{\frac {dv_{1}}{dt}},{\frac {dv_{2}}{dt}},{\frac {dv_{3}}{dt}},\cdots \right]\ .

{\ displaystyle {\ frac {d {\ vec {V}}} {dt}} = \ left [{\ frac {dv_ {1}} {dt}}, {\ frac {dv_ {2}} {dt} }, {\ frac {dv_ {3}} {dt}}, \ cdots \ right] \.}

Physics application

Time derivatives are one of the key concepts in physics. For example, for a radius vector $x$ ${\ displaystyle x}$ time derivative ${\dot {x}}$ ${\ displaystyle {\ dot {x}}}$ this is his speed , and the second time derivative ${\ddot {x}}$ ${\ displaystyle {\ ddot {x}}}$ this is his acceleration . The third time derivative is known as a jerk .

A large number of equations in physics is a time derivative of a vector, such as velocity or displacement. Many other fundamental values in science correlate as time derivatives of each other:

force is time derivative of momentum
power is time derivative of energy
electric current is time derivative of electric charge

Economic applications

In economics, many theoretical models of the evolution of various economic variables use time derivatives.

Notes

Iang Chiang, Alpha C., Fundamental Methods of Mathematical Economics , McGraw-Hill, third edition, 1984, ch. 14, 15, 18.

[1] Iang Chiang, Alpha C., Fundamental Methods of Mathematical Economics , McGraw-Hill, third edition, 1984, ch. 14, 15, 18.