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!