vysero said:
So the d in dx/dt does not stand for derivative does it, it stands for differentiate?
Ohh boy, okay. We need to backtrack a little. Let's talk a bit about terminology and notation:
1. To "differentiate" a function means to
take the derivative of that function. So we say "differentiate", we DON'T say "derive" or "derivate." So the
action of computing a derivative is called "differentiating." Similarly, "differentiation" is the noun for that action or process.
2. There are several different ways to denote a derivative. I.e. there are a bunch of different mathematical notations.
a. The notation of Newton, which is to use a prime symbol (##\prime##) next to the function, to indicate that it has been differentiated once. So, if you have a function ##f(t)##, then the first derivative of that function with respect to t would be denoted as ##f^\prime(t)##. You add another prime symbol for the second derivative: ##f^{\prime\prime}(t)##. Note that
in physics, we often don't bother introducing a second letter for the function notation. So, in your example, instead of saying ##x = f(t)##, we would just write the function as ##x(t)##, which indicates in a compact way that x is a function of t. Then the derivative would be ##x^\prime(t)##.b. The notation of Leibnitz, which is as follows: if we have a function x = f(t), then the derivative of that function with respect to time is given by the symbol:$$\frac{df(t)}{dt}$$which can also be written as$$\frac{d}{dt}f(t)$$ The "d" just by itself doesn't mean anything. The entire expression ##\frac{d}{dt}## should be regarded as ONE symbol that
acts on the function. (It's NOT a fraction or a division of two things) In fact, when we have something that
acts on a function to carry out some sort of mathematical
operation on it, we call that an
operator. So the symbol ##\frac{d}{dt}## can be regarded as the "differential operator". In other words, it is the operator that carries out the operation of differentiation with respect to time. Any function that you put next to this operator is being differentiated with respect to time. That's why the second way of writing it: ##\frac{d}{dt} f(t)## is more suggestive, because it makes it clearer that the differential operator d/dt is acting on the function f(t) to produce the derivative of f(t) with respect to time.
These two notations MEAN the same thing. Both of them are ways of writing a derivative. I.e.$$f^\prime(t) = \frac{df(t)}{dt}$$
As I mentioned before, physicists are lazy/sloppy with notation. So, OFTEN in the Leibnitz notation, the argument of the function is omitted altogether, and we just write ##\frac{df}{dt}##. Furthermore, just like I said above for the Newton notation, we sometimes don't bother with the 'f' at all, and just write the function using same symbol as its variable: x(t). In this case, for the Leibnitz notation, the derivative would be written as ##\frac{dx}{dt}##. In fact, with the Leibnitz notation, the letter for the dependent variable (y, or in this case x) is what appears more often than the letter for the function (f or g or whatever), assuming the two are even distinct. I.e. if y = f(x), then in the Leibnitz notation, you'd might more often write dy/dx, rather than df/dx, for the derivative.
Again, I want to emphasize that although dx/dt (or df/dt) looks like a ratio of two quantities, it is not. It's just a symbol for "derivative of x with respect to t." So why did Leibnitz choose to write it that way? To be honest I don't know, but to me, I think it is meant to be suggestive. For, although dx/dt is NOT a ratio, the way you compute is by causing a small change Δt, and seeing what the resulting change Δx is. The ratio of these two Δx/Δt is the
average rate of change of the function x(t) over the interval Δt. To figure out the instantaneous rate of change of x, i.e. how fast x is changing at particular instant in time, you take the LIMIT of this ratio Δx/Δt as Δt → 0 (as you are no doubt learning in your math class). This *limit* is the formal definition of the derivative of the function: dx/dt. So although dx/dt is not a ratio, its value is equal to the
limit of the ratio Δx/Δt.
Again physicists are a bit sloppy and DO tend to think of dx/dt as a ratio of an "infinitesimal" (i.e. vanishingly small) change in position dx to an "infinitesimal" change in time dt (as opposed to Δx and Δt, which were
finite changes in those quantities) So they DO treat dx/dt as a fraction, and sometimes even separate the dx and the dt, even though this is not strictly correct and is an "abuse" of notation. But I mention it because you will no doubt see it in your physics classes. The reason why it is frowned upon somewhat is because, at least at the intro calculus level, we have no precise way of defining what an "infinitesimally small" quantity is. So this is not a well-defined concept. Instead, we define the derivative using the concept of a limit, which IS a precisely-defined concept in mathematics.
