OH SHIT IT’S LEIBNIZ NOTATION.
The importance of good notation is often overlooked. Sometimes, in your mathematical journeys, you may think “Why the balls are they making a new symbol for this idea I already understand?” There’s a good reason for this – just like in good writing or speaking, in good math, clarity is important. Any time you want to say something in math, you’re making a choice as to how to express it.
As we’ve discussed in the past, the derivative is a certain operation you do on a function. More specifically, it’s a particular type of limit that is super useful for describing reality. It’s so commonly used, that putting an apostrophe next to a function denotes differentiation. However, there’s another notation, which was introduced by Leibniz (pictured below), which I think is very good for illustrating what the derivative is actually doing.
SO, recall that when we say “y” we’re typically referring to some function of x. So, y is the same as f(x).
Also recall that when we take a derivative of a function of x with respect to x, we’re saying “how does that function of x change as x changes?” So there are two things there: the function and the derivative. For the function, we’ll just write “y.” To ask that question about change, we’ll write ”d/dx.” So, the whole expression ends up as “dy/dx,” which is typically pronounced “dee y dee x.”
You will encounter “dy/dx” a lot as we go through this book, so it’s important you deeply understand it. I think many students think “dy/dx” is just some made up notation that you write after taking the derivative of y. It’s not. It’s an application of the mathematical move “d/dx” to the function “y.” The “d/dx” means “how does SOME FUNCTION change when x changes.” The “y” is that function.
I hope I haven’t belabored this too much, but if you don’t understand what you’re doing when you write dy/dx, you’ll get really messed up later on.
Additionally, there’s another bit of notation that’s handy, and looks like this:
The bar there means that you’re just evaluating f(x) at a certain point denoted by a. If you don’t have that bar notation, you’re just taking the general derivative of a function. If you do have that bar, you’re finding a value of the derivative at a certain point. In other words, without that notation you’re saying “how does y change with x.” With it,, you’re saying “how does y change with x at a.”
Another way to denote that would be f ‘(a). Makes sense, right? You’re doing the following steps: 1) Take the derivative of a function to create a new function. 2) Find the value of that new function at a.
This leads us to an important question: Are functions differentiable at every point?
Stay tuned to find out in the next episode of… Calculus!
Shouldn’t all first year calc students know this already? If not, they better retake that class.