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Physics problems are usually written like:

The rate of change of the soup's temperature ...

Is there a common English word that captures "rate of change" or "speed of change" in a single word, other than derivative?

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That's hard, because phrases like "rate of change" have such specific technical meanings that something will invariably be lost in their substitution. Another example is "goodness of fit". I would argue that "derivative" is actually not a good synonym for "rate of change" because it denotes the mathematical operation associated with a "rate of change", but not the notion of "rate of change" itself. – Gilead Feb 14 '11 at 0:14
Are you looking for a word that is valid in a specific context, or in general? – kiamlaluno Feb 14 '11 at 0:16
However, if you are looking for synonyms for the mathematical idea of a derivative, there are: 1) differential coefficient; 2) gradient/slope function; or simply, 3) differential. – Gilead Feb 14 '11 at 0:18
@Gilead - I provided gradient below. Slope is also a synonym for "rate of change" but I couldn't imagine using the word slope in the context of soup:-) – ukayer Feb 14 '11 at 1:12

5 Answers

up vote 5 down vote accepted

Speed is used for the rate of change of distance with respect to time. Sometimes speed is used in contexts similar to what you mention.

How quickly will the soup reach room temperature.

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I'm settling for "growth speed", "motion speed", and "temperature speed". Its a bit awkward, but I like how speed is scalar and doesn't imply direction, leaving the quantity that is changing to determine direction (negative growth=shrinkage, negative motion=backwards, negative temp=cooling) – bobobobo Feb 14 '11 at 12:17
Sorry, that should have been heat speed, not temperature speed, since temperature is not a verb – bobobobo Feb 14 '11 at 13:07
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@bobobobo - heat speed doesn't really work, even in your example sentence, "the heat speed of the soup..." would be quite wrong. How are you planning to use the term? I should add that @David's rephrasing is just fine if that is what you are looking for. – ukayer Feb 14 '11 at 14:50
@bobobobo "growth rate" -> growth rate, "motion speed" -> speed, and "temperature speed" -> warming rate. This is because growth already implies delta size vs. delta time; warming implies delta Temp vs. delta time. – David W Feb 14 '11 at 14:55
I am purposefully using the same word to mean "rate of change" - even if it does not quite seem to fit at first. Can you point out what is wrong with heat speed, except that it is not in common use (yet? ;) I'm talking about the rate of change of the temperature of the soup - in other words, the heating speed of the soup. If it is +, then the soup is heating, if it is -, the soup is cooling. – bobobobo Feb 15 '11 at 4:24
up vote 4 down vote

I tend to use 'delta', but it's what I call an 'acquired' definition - it's one I picked up along the way, but I have no idea if it's the correct one.

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Differential or delta is what I would use, but only with scientists and mathematicians. Otherwise I would always say "rate of change". – Orbling Feb 14 '11 at 0:32
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Delta technically denotes "change" though, not "rate of change". In a difference quotient, the ratio of two deltas give a rate of change. – Gilead Feb 14 '11 at 0:39
I think saying "delta f" or "the delta of f" or "the difference of f" is fine if f is sampled. The denominator in the quotient is the trivial "1 sample", and it's easily inferred. That said, it's the equivalent of "the derivative of f with respect to" in the continuous case, so I think, as per the question's exclusion of "derivative", it's not really the answer. All I could think of was Newton's "fluxion", and that fails for being uncommon. – eryksun Feb 14 '11 at 10:23
up vote 3 down vote

Another synonym is "velocity". In pediatrics they say "height velocity" to refer to the growth in stature per year. For the second derivative you can say "acceleration".

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Good answer. Going with speed, it should not imply a direction – bobobobo Feb 14 '11 at 12:18
up vote 2 down vote

You could use gradient for the example given, e.g. "the soup was being warmed with a temperature gradient of 10 degrees every 5 minutes".

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That's not a typical use of gradient - usually it means the differential in physical space... – HorusKol Feb 14 '11 at 0:13
@HorusKol - I agree. The only phrase I personally would ever use is "rate of change". – ukayer Feb 14 '11 at 1:08
The NOAD reports that the meaning of gradient is an increase or decrease in the magnitude of a property (e.g., temperature, pressure, or concentration) observed in passing from one point or moment to another. – kiamlaluno Feb 14 '11 at 1:20
@kiamlaluno - exactly, when you look at the gradient or slope of how a variable changes over time you are measuring the increase or decrease of the variable as you pass from one point or moment to another. Temperature gradient is perfectly correct. – ukayer Feb 14 '11 at 1:24
@ukayer: I was referring to the first comment of HorusKol. Clearly in physics, gradient refers to the change between different points in the space; as modern physics introduced the concept of four-dimensional space, even a change between two different moments is a change that happens in two different points of the four-dimensional space. I am sorry for the confusion: I should have said to whom was directed the comment. – kiamlaluno Feb 14 '11 at 1:31
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up vote 1 down vote

@HorusKol: There you go. Differential is the right word.

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I'm not sure about that. Consider the OP's sample sentence. If you had substituted "differential" into it, it would read: "The differential of the soup's temperature...", a substitution which does not seem to be correct in the given context. – Gilead Feb 14 '11 at 0:38
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Differential doesn't imply the rate of change with respect to time the same way that speed does. – David W Feb 14 '11 at 1:02
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@Yitzchak - did you mean to comment rather than answer? Differential does not mean rate of change. – ukayer Feb 14 '11 at 1:10
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Unfortunately, the 'differential of a temperature measurement' would actually refer to the change in temperature, and not the rate of change. – HorusKol Feb 14 '11 at 2:35
Yeah I meant to comment. @Gilead: Consider this structure of the sample sentence: "The soup's temperature differential..." – Yitzchak Feb 15 '11 at 2:58
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