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?
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.
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.
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".
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".
If you were an economist you might talk about the "marginal increase in temperature of the soup". Outside economics the word is not used very much in that sense.