This is the graph of the line y = x:

If you put your finger down on any point on that line, and then put another finger on another point on that line, you find that the total change in the y-coordinate divided by the total change in the x-coordinate between those two positions is 1. Move two units to the right, and the line rises by two units, etc. This is the same no matter which two points you pick. Every line has this sort of property, which is called *slope*. Steep lines that rise greatly for each unit of x displacement have a large slope, flat horizontal lines have zero slope, and descending lines have negative slope. In each case the way to calculate the number describing the slope is just to divide the amount of vertical rise by the amount of horizontal run. If you have the equation of the line, the slope is just the number in front of the x. I.e., y = 3x has a slope of 3, because if you increase x by some amount, y increases by three times that amount.

But what if your equation isn’t a line? Here’s y = x^2:

Clearly this curve hasn’t got a constant slope. But this doesn’t stop us from at least informally describing each point on the curve as if it had a slope. Over on the left, the slope seems to be negative, since a little ball places on the curve would be sliding downward. Then it gets less and less negative, until at the bottom of the parabola the slope seems to be zero. Then the slope becomes more and more positive as the graph rises more and more for each bit of distance run.

That’s pretty informal though. We might want to assign a number to that ever-changing slope, and we need a good way to define that slope. I propose we define the “slope of a curve at a particular point” as the slope of the line (which is well defined) which happens to be tangent – just touching – the curve at that point. For instance, if we wanted to find the slope at x = 1, we’d look at the slope of the line touching the graph at that point:

Which is nice, but right now we have no way of knowing what that line is. To write the equation of a line, you either need a point on that line and a slope, or two points on that line. We only know one point, and we’re trying to find the slope. Can we get around this? Well, we can try to do so by putting one point where we’re trying to find the slope and one point farther to the right by a distance delta x. Then we can shrink delta x and see what happens to the slope. Since the slope is the change in y over the change in x, we’ll just call it Δy/Δx.

Well, it looks like the slope as x = 1 is about 2. It sure seems to be closing in on that value anyway. But we want to be sure of this. We’ll calculate it by hand for a generic x, though in this case of course x = 1 which we can substitute in at the end if we want.

The change in y (which we called Δy) is easy to write down. The change between two numbers is the end value minus the start value. The start height for a given x is just x^2, and the end height will be the square of x plus however far we went to the right:

Therefore the slope between the points x and (x + Δx) is:

We can go ahead and multiply out the top:

Cancel the x^2 and divide by delta x:

But the whole point is that we want the change in x to approach zero. This means the only surviving term is 2x. And that’s the slope of the tangent line to the parabola y = x^2 at any point x. It’s our Sunday Function, and for its official presentation I’ll switch the deltas on the left side into the d-notation favored in modern calculus:

The notation dy/dx is the standard way of expressing the statement “the slope of the tangent line to the function y at a given point x”, which takes up way too much space to write down. “d” is just part of the notation, it’s not a new variable or anything.

And that’s one half of the basic concepts of calculus – finding the slope (or more generally the *rate of change*) of a function at a given point.

The eagle-eyed among you may not be happy with this. You might say something like this: “If delta x is greater than zero, you can’t really be said to have truly found rate of change at exactly one point. But if delta x is actually equally to zero, then you’ve divided by zero, which is impossible. Either way this method doesn’t quite work.”

And you’d be right. But calculus did nonetheless work perfectly for solving problems (heck, we just solved a problem with it), so for a while mathematicians and scientists were more or less willing to press on and keep developing the calculus in the anticipation that a more formally careful method of finding slopes could be found, avoiding the divide-by-zero problem. Sure enough, the 19th century mathematicians Augustin-Louis Cauchy and Karl Weierstrass came up with a formally correct if conceptually recondite way to re-express this procedure without division by zero. With calculus on a firm logical footing, this process of *differentiation* is ubiquitous in modern science and engineering.