Calculus was developed as a solution to a very specific problem: the motion of objects through space. Though its applications range far beyond that problem, that original problem remains by far the best way to approach calculus since everybody already has intuition and experience with moving objects.
In a better world, then, calculus will always be approached from a physical perspective, since everyone already sort of understands how things move around. Here's how. [This is, incidentally, what I did in the first day of AP physics class, but as you'll see, it's not terribly complicated, and (hopefully) most anyone can follow it.]
Imagine you're standing around with a stopwatch on a road that conveniently has length measurements posted all along it, and a car drives past you. Your task is to measure how fast the car is going the instant it passes the mark that's right at your feet. How do you do it?
Well, speed is just a measure of how far the car goes in some unit of time, say, a second, so you can just start your watch as the front wheels pass the mark by your feet, and then mark off where the front wheels of the car are when the watch reads exactly one second. (We can ignore the fact that, perceptually, this might actually be a difficult task...imagine you have some helpers or something). Let's say it's gone 10 meters, as marked on the road. Then it's speed is just 10 meters / 1 second=10 meters per second. Right?
Almost. What you've measured is the car's average speed over one whole second. But remember we want to find the speed of the car the instant it passes by your feet. Let's say it passed you by quite slowly but then managed to speed up incredibly quickly and travel 100 m by the time your stopwatch reached the one second mark. You wouldn't conclude that it was going 100 meters/sec when it passed you.
So, you say, okay, let's not measure the distance it travels in a whole second after it passes me, as it can speed up, slow down, and do all sorts of crazy things in that time! Let's measure the distance it goes in just a tenth of a second!
This approach will have the same problem, but at the same time, it's definitely getting us closer to what we want. The car can speed up or slow down in a tenth of a second just as it can speed up or slow down in a whole second, but it can't speed up as much. What you'll end up measuring is, again, the average speed of the car over one tenth of a second. But that's probably closer to the speed we're looking for.
Okay, so make it a hundredth of a second, or a thousandth! Okay, you're getting the idea. No matter how small you make the time interval over which you're measuring, the car will always move some finite distance over that time interval. You can basically think of the speed as the distance you travel in some tiny time interval, divided by the time interval. If the car goes 10 millionths of a meter in 1 millionth of a second, then it's speed is very well approximated by .000001 meters/.0000001 seconds=10 meters per second.
[Now, if you want to be more precise, the above definition of speed doesn't quite cut it (but it's close enough, so you can probably skip this paragraph). Really, you take all these different tiny time intervals, say a thousandth, a millionth, a billionth, and a trillionth of a second, and mark off where the car is after each time interval. You find the average speed associated with each time interval as we did above for one second and one tenth of a second. If they're the same, great! You're done. That's your speed. But even if they're different, you'll notice that as you make the time interval smaller and smaller and smaller, the speed you calculate will get closer and closer to some value. That's the speed.]
Congratulations, you now more or less understand the idea behind the derivative—one of calculus's two essential ideas! In this case, what we were looking for was the speed. But here's how we found it: we took the change in position (how far the car moves) and divided by the time interval, meanwhile shrinking the time interval so that it was arbitrarily small. In math jargon, this looks like
where x stands for position, t stands for time, and the Greek letter delta means "change in." This, then, we re-define as the derivative of position with respect to time. We solved our problem, and we generalized our solution to a definition, which will be very useful later on!
In the next post, I'll discuss the standard approach to calculus a little more thoroughly.