Basic Differentiation For Anyone Who Has Had Algebra I
With Pictures!
And Without Formulas!

Calculus is really not that hard; it just sounds intimidating. So I am going to talk about basic calculus, with differentiation and integration. Just basics, mind you, because for the slightly more advanced stuff you do need some higher math than the stuff you learned in Algebra I. This will just be a basic explanation of the concepts behind differentiation, without any formulas. See, to get to the formulas like the Power Rule and so on, I have to talk about limits, and I don't really want to do that. So this will just be an explanation of the concepts.

Everyone who is here should know what a slope is. Rise over run. I'm sure that, if your math teachers were anything like my math teachers, they made you learn that about fifty billion times. So we can find equations of simple lines like the one in this picture:



This is supposed to be the graph of the line y=x, although it may not be exactly to scale. If you know the formula of a line-- y=mx+b, where m is the slope and b is the y-intercept-- you should be able to tell that this line has a y-intercept of 0 and a slope of 1. In other words, for each unit you go up on the graph, you also go over one unit. Rise/run = 1/1. Thus, the point (0,0) is on the line-- as you can easily see on the graph-- and so are the points (1,1), (2,2), (3,3), and so on. I didn't label the points on the graph because I'm lazy, so the graph may not be exactly to scale, but you get the point. This is easy to see, because the graph is of a straight line; everywhere on that line, the slope has the same value, which is 1. So if we were to plot the value of the slope of this graph at each x-value, it would look something like this:



In other words, no matter what x-value you look at the slope at, it will always be 1. At the point (1,1) the slope will be 1. At the point (2,2) the slope will be 1. And so on, for each point on the line y=x. So we graph those values at each point, as shown in the diagram above.

But what if you have a graph like this one?



Obviously, this is not a straight line. In fact, it's supposed to be the graph of the equation y=x^2. (Again, it's probably not to scale.) You can see that the bigger x gets, the steeper the slope of the graph gets, or the bigger the value of the rise compared to the run gets. In fact, here's a picture of the graph with some sample slopes drawn in.



How did I decide what the value of those slopes would be, and how did I know that they would get bigger as x got bigger? Pretend that you can zoom in on any part of the graph drawn above there. In fact, you can zoom in so far that when you look at a part of the graph, it looks just like a straight line. You can think of this curvy graph as being composed of lots and lots of little straight lines. Each little straight line corresponds to an x-value on the graph of y=x^2. See the picture below for an illustration of what zooming in on the graph would look like.



To use an analogy, think about this: the Earth is round, right? But where you're standing/sitting, the surface of the Earth probably looks reasonably flat. You can think of the part of the Earth where you're standing as a really tiny part of the entire surface of the Earth. So even though the Earth is round, the piece of it that you're standing on is flat. You can put lots of those little flat pieces together to make something that's shaped like the Earth. A curve is also like that: you can put lots of little straight lines together in the shape of a curve. Each of those little straight lines has its own slope, and if you think of the curve as being composed of little straight lines, then you can easily see that the slopes of those straight lines get bigger as the x-values on the graph of y=x^2 increase. In the illustration where the slopes are drawn in color, I exaggerated the size of the straight line in order to show their slopes. We call those exaggerated straight lines tangents to the curve. I linked the Wikipedia entry on tangents there, because I don't want to get too sidetracked by discussing what a tangent is. Just read the "Geometry" part of that entry, because that's the relevant part.

Each of those straight lines, as I said, has its own slope. And in the case of this equation, y=x^2, as the x-values get bigger, the slope also gets bigger. We might guess that therefore, the slope of each little line segment is linked to a corresponding x-value. We would be right, but let's not go there right now. As I said before, each little straight line is linked to an x-value. But these x-values aren't necessarily integers. In other words, there's a little line segment with its own slope at x=1. But there's also a little line segment at x=1.2, and x=2.5435423, and at every conceivable (real) x-value you could think of that's on the graph of y=x^2. There are enough little line segments to make that curve look pretty smooth when we look at it like in the diagrams above.

Say what? Well, then how are we supposed to tell which slope values we're supposed to be looking at? At which x-values should we find the slope of the corresponding little line segment? Should we be graphing the slope of the little line segment at x=0, or x=1.2, or what?

This is where the slightly counterintuitive part comes in. The trick is to think of the curve, y=x^2 in this example, as having infinitely many little line segments. Each infinitely small line segment also corresponds to an infinitely small x-value. Think of the x-axis of our graph as being divided up into infinite little pieces, with each infinite little piece corresponding to an infinitely small line segment. Remember that each of these little line segments, when put together, will make up the curve y=x^2 that we see in those graphs. In calculus, we call each infinite little piece of the x-axis dx. This will come in handy when we learn about integration.

Here's something to watch out for: just because we divided the x-axis into an infinite amount of pieces, that doesn't mean that each portion of the x-axis we look at will be infinitely long. For example, although there are an infinite amount of dxs in the space from x=1 to x=4, that doesn't mean that the space from x=1 to x=4 is infinite. In fact, the space from x=1 to x=4 has a length of 3. See, as it turns out, you can add an infinite amount of things together and come out with a finite number. Does that seem counterintuitive? IT IS. But it's true. For an example, try adding up this sequence of numbers: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^x)+ ... (Here, the value of x goes from 0 to infinity, since this sequence of numbers is infinitely long. So we add up 1/(2^0) + 1/(2^1) + 1/(2^2) + 1/(2^3) and so on.) Each number you add to the sequence is smaller than the one that came before it. So, while the number we get by adding up infinitely many terms of this sequence keeps getting bigger, it gets bigger by a smaller number each time. It makes intuitive sense that, therefore, the number we get by adding up all of these successively smaller numbers isn't going to be infinite, but add up to a single number. (It adds up to 2, in case you were curious.) Furthermore, each value we add successively is small enough so that the sequence doesn't equal infinity when you add it all together. In other words, the numbers get small quickly enough so that the sum of this sequence is a single number. For comparison, the sequence 1+1/2+1/3+1/4+...+1/(x)+... equals infinity when you add all those numbers together. The values that you keep adding on don't get small enough quickly enough to add up to a single number.

This may not make any sense, but if you don't quite get it, that's fine. All you need to really understand is that we're dividing both the curve and the x-axis into an infinite number of really small line segments. And we call each piece of the x-axis dx.

Calculus is sloppy in this regard: it doesn't bother to come up with a precise definition for exactly how big dx, that infinitely small piece of the x-axis, is. We just think of it as being "infinitely small." If you can understand this, you'll be a pro at integration, which I'm not discussing in this essay because it's quite long enough already.

Now that you hopefully understand what dx is, let's return to the curve of the equation y=x^2. We've established that there are infinitely many small line segments that make up the curve y=x^2. (Well, actually, we haven't established anything, because I haven't given any mathematical proofs, only logic and analogies. But just take my word for it.) A derivative is the term for the slope of each infinitely small little line segment. In other words, it's the slope of the curve-- in this case, the curve is y=x^2-- at each dx. On some graphs, like the line y=x, the derivative will be the same at each point, each dx. On others, like the curve y=x^2, it will depend on the x-value. We call the process of finding the slope of each little line segment differentiation. The slope of each little line segment is still rise over run, though. However, in this case, we use that tiny little piece of the x-axis-- dx-- as our "run." You might guess that, by analogy, we also use a tiny little piece of the y-axis, which we call dy, as the "rise."

We can also plot the slope of the curve y=x^2 at each x-value like we did with the line y=x (see graph above). However, as we've noted, the slope of this curve seems to depend on the x-value, and is different for each x-value. (This doesn't hold true for all curves-- in other words, on some curves, some x-values will give the same slope-- but it does hold true for the curve y=x^2.) Therefore, the graph of the slope vs. x-value won't be a flat line, like it was for y=x.

You can think of the slope of a graph, or its derivative, as the rate of change of the value that's graphed. For example, let's say that we're graphing the height of a ball thrown up into the air vs. time. The graph might look something like this:



The ball will go up in the air until it reaches some maximum height and then come down. Remember that a slope is rise over run. If we look at the units of each of those values-- height and time-- and remember that a slope is rise over run, we can see that the unit of the slope of this graph is (unit of height)/(unit of time). This might be, say, feet/second. In other words, the slope is the rate at which the height of the ball is changing.

Lastly, there is also a concept called the second derivative. Put simply, this is the derivative of the derivative. I'm just going to explain this briefly because the point of this essay was to explain basic concepts behind derivatives. You can see how, in the graph of y=x^2, the slope at each point/dx seems to depend on the x-value; it gets bigger depending on what the x-value is. I already pointed this out. If we use the example of the ball being thrown up in the air, we can therefore not only find the rate of change of the height of the ball, but also the rate at which that is changing. The ball's rate of change of height may be changing by 2 ft/second each second, for example. So at one point, say at 2 seconds, the ball's height may be changing by 4 ft/second. But at 3 seconds, the ball's height may be changing by 2 ft/second, since we've said that the ball's rate of change of height is changing by 2 ft/second^2, or 2 ft/second per second. So the second derivative is the rate of change of the rate of change of the ball's height. I hope that that makes sense.

There are lots of concepts that are built off this understanding of derivatives. For example, if you understand what a derivative is, you can find, very precisely, what the maximum and minimum values of a function are. But those applications are a topic for another day; I'm just trying to explain what a derivative is here.

That's it. Now you understand some basic calculus.

If you actually understand this, drop me an email. For that matter, if this isn't clear enough, drop me an email too and tell me which parts aren't clear.