To compare masses, we use a balance, or a scale. To compare lengths, we use a ruler or a tape. This is a familiar procedure to us all, even if we often lose sight of this fact while doing calculus homework. In typical college calculus courses, numbers are manipulated and related to each other with sparingly little discussion of what the numbers are intended to represent. Of course, the whole purpose of numbers in the first place is to represent the characteristics of physical things .
Physicists are more careful about keeping track of physical units.
When modern engineering students study basic newtonian physics, they are quite often cautioned to include the units of measure in their answers, if not to go so far as to carry the units all the way through their calculations. In chemistry, the instructors are even more emphatic. Many problems in chemistry amount to little more than units conversion. The basic idea of calculus, that is the thing that defines it and separates it from algebra, trigonometry or analytic geometry or any other mathematics, is the notion of an infinitely small, discrete entity. Differentiation is a process wherein two infinitely small differences are divided by one another, whereas integration is a summation of an infinite number of infinitely small differences.