This particular method of writing derivatives is Leibniz's notation, there is some interesting information of information on Wikipedia about this type of notation.
Hope this helps
I understand P'(t) alot easier, but I'd like to understand the other notation. (p = position, t = time, v = velocity, a = acceleration).
My first question.. is this an actual fraction? What's in the numerator, and what's in the denominator?
Is the input variable necessary in the top? Could it just be:
Now, about stacking derivatives:
Ok.. are these actual exponents or are they just notations to indicate how many derivatives have been done? And why is it that the "exponent" is placed on d in the numerator and placed on t in the denominator?
No, it is not an "actual fraction"- it is defined exactly like p'(t) is: . However, it has a nice property: you can go back before the limit, use the fraction property, and then take the limit so all "fraction properties" hold! You can define "differentials": "dx" is simply symbolic and we define "dp= p' dx" to make use of the fact that, although dp/dx= p' is NOT a fraction it can be treated like one! That's a very "vague" definition. "Differentials" are defined more precisely and used a lot in higher mathematics like Differential Geometry.
Yes, if it is understood what the variable is. That's commonly done. And, of course, we first define the derivative at a specific point, with the derivative function being defined as the function that takes on those values at each point. If I want to talk about the derivative of p(t) at t= 3, I would have to writeIs the input variable necessary in the top? Could it just be:
No, those are not exponents. It is also true that higher derivatives like that [b]cannot[\b] be treated as fractions. The position of those numbers is just custom. One reason, perhaps, is to remind us that they CANNOT be cannot be treated like a fraction.Now, about stacking derivatives:
Ok.. are these actual exponents or are they just notations to indicate how many derivatives have been done? And why is it that the "exponent" is placed on d in the numerator and placed on t in the denominator?