# Rate of change

The rate of change of a time-dependent variable describes the extent of the change in over a certain period of time in relation to the duration of this period. In vivid terms, it is a measure of how quickly the size changes. By referring to the duration, the unit of measurement in the denominator contains a unit of time; in the numerator there is a unit of . If the change is also related to the size itself, one speaks of a relative rate of change or growth . ${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle G}$

A distinction is also made between the mean rate of change between two measurements and the current (also local ) rate of change as an abstract quantity of a model.

## Calculation and use

### Medium rate of change

The mean rate of change is the average change in a time-dependent measured variable between two points in time and , that is, in the period . It is calculated as the quotient of the difference between the two values ​​at these points in time and the duration of the period:${\ displaystyle G}$${\ displaystyle t_ {1}}$${\ displaystyle t_ {2}}$${\ displaystyle \ Delta t = t_ {2} -t_ {1}}$${\ displaystyle \ Delta G = G (t_ {2}) - G (t_ {1})}$${\ displaystyle \ Delta t}$${\ displaystyle {\ tfrac {\ Delta G} {\ Delta t}}}$

In the time-quantity diagram ( function graph, diagram) of , the mean rate of change between and is the slope of the secant through the points and on the diagram. ${\ displaystyle G (t)}$${\ displaystyle t_ {1}}$${\ displaystyle t_ {2}}$${\ displaystyle (t_ {1} | G (t_ {1}))}$${\ displaystyle (t_ {2} | G (t_ {2}))}$

### Current rate of change

The current rate of change is the change in a measured variable related to a “moment” (very short period of time) . It can be mathematical as a result of the boundary process ${\ displaystyle G}$

${\ displaystyle {\ frac {\ mathrm {d} G} {\ mathrm {d} t}} = \ lim _ {\ Delta t \ to 0} {\ frac {\ Delta G} {\ Delta t}} = \ lim _ {\ Delta t \ to 0} {\ frac {G (t + \ Delta t) -G (t)} {\ Delta t}}}$

can be represented as a derivative of their time function . ${\ displaystyle G '(t)}$${\ displaystyle G}$${\ displaystyle G (t)}$

For time-linear changes, the instantaneous rate of change is constantly equal to the mean rate of change.

### Rates of change in a broader sense

If the terms are used figuratively for quantities that depend on a parameter other than time, then: ${\ displaystyle G (q)}$${\ displaystyle q}$

• the mean rate of change is equivalent to the difference quotient ${\ displaystyle {\ tfrac {\ Delta G} {\ Delta q}}}$
• the current rate of change is equivalent to the differential quotient ${\ displaystyle {\ tfrac {\ mathrm {d} G} {\ mathrm {d} q}}}$

If the parameter is a vector quantity, the term “ gradient ” is also used instead of the term “rate” , such as temperature gradient or air pressure gradient . ${\ displaystyle q}$

## Examples

• In the case of a linear movement, the speed is the current rate of change of the time-distance function . The article speed in the section definition of speed makes the difference between mean and instantaneous rate of change clear.${\ displaystyle v (t)}$${\ displaystyle x (t)}$

## literature

• Harro Heuser: Textbook of Analysis Part 1 . 5th edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2
• Christian Gerthsen, Hans O. Kneser, Helmut Vogel: Physics: a textbook for use alongside lectures . 16th edition. Springer-Verlag, 1992, ISBN 3-540-51196-2

## Remarks

1. Helga Lohöfer: Table of the usual change terms for variables and functions. Script for the exercise Mathematical and Statistical Methods for Pharmacists , University of Marburg. 2006.