Yeehaw! We’re one section away from getting into the nitty gritty. But, this last section of chapter 2 is very important. This is where you stop thinking of the derivative as a trick and start thinking of it as a function!
To reiterate once more: A function is how you map from one set to another. You have a set called x. You do an operation called f(). That gives you a new set: f(x). You can also change the f() to map from x to a different set. Some ways you can do this are trivial. For example, you might have a function f(x) = 1 that maps from all numbers to 1. Not very exciting. You can then change it to f(x) = 1 + x. Now you’re mapping from every real number to every real number plus 1. You could change it again to f(x) = 1 + x + and get further adjustments still.
The derivative is just another way to map. Here’s it’s definition:
This is the way you map from x to f’(x), aka the derivative of f at x.
The next part of this section is about developing your graphical intuition. This can be a little mindscrewing, but it’s not so bad once you get it. Recall that the derivative at a point is graphically the same as a tangent at that point. So, if you start with graph A, then make a new graph (graph B) built of graph A’s tangents, you’ll find that B is the derivative of A.
Here’s a simple example: Say you have a function, f(x) = 2x. You should quickly see that as the equation of a line. Specifically it’s a line that increases 2 vertical ones whenever it increases 1 horizontal unit. So, the tangent’s pretty easy here – remember the tangent is just the slope at a point. This function has the same slope everywhere: 2. So, if we take the tangent anywhere, we find it’s a line with a slope of 2. So, we know that the derivative of f(x) = 2x is simply 2. That is, f’(x) = 2.
f’(x) is thus very easy to plot. It’s a constant function that’s always at 2. By now, you may see the logical extension. Any linear function’s derivative is just the coefficient of the variable. Nice!
With more complicated functions, it gets… well… more complicated. We’ll save that for later chapters. For now, try drawing out a random curve on a graph and see if you can figure out the derivative. It can often be very counterintuitive. But, if you remember rules like the one above, for lines, it can help. For instance, anywhere on your random curve that has a roughly constant slope should have a roughly horizontal line for its derivative. If the slope is harshly up, that constant is a high value. If it’s sharply down, it’s a low value. I encourage you to play around and try to find more such tricks.
The book also has you find some derivatives using the equation that’s higher in this blog post. To be honest, I’m not sure how much utility that is. It’s probably good in the way your dad making you shovel snow out of the driveway is good – builds character. But, you’ll soon learn much simpler ways to deal with these problems.
Next section: Leibniz notation.