Okay:

A function is a mathematical expression where we take x values, plug them into the expression and get y values back, right? So for instance, the function "y = x^2" means that for each x value, if you square it, you get a y value back.

You can use functions to make graphs, by performing the computation of the function's value and then plotting the points. The points are defined like (x , y) coordinates, so the x value is first and then the y value returned by the function then comes.

Like, with our y = x² example, our first point would be (1, 1) , then the next would be (2, 4), then (3, 9) and so on. All of the points in between would be calculated the same way. This is what such a graph would look like. You can try other functions, and it should become clear what a given function will produce as graphed output.

Now, with functions, you can see that it's possible for there to be what's called a "tangent line", in other words, it's a line that touches the graph of that function we just plotted in only one place. (You can remember the term because "tangent" is latin for "touches") Here's a really good illustration.

Here's what the derivative is: It is a function, that is derived from a first function that tells you what the slope of the tangent line is at each point.

Here's an example. We have the original function y = x^2. If you use rules for deriving, you end up with a derivative (That's the name of the function) of y = 2x . What this means is that, at each point on that original graph, the slope of the tangent line is equal to 2 times the current x coordinate you're on.

If you were following y = x² with a pencil and making the pencil a tangent line, at (1, 1) of the original function, the slope of the pencil would equal 2. At (2, 4), the slope would equal 4. At (3, 9), it would equal 6.

(There are rules for differentiation in text books and on the internet. Understanding the concept of the derivative is more important than knowing the rules if you're just trying to get the big picture. I've left out how I derived the function, which functions are differentiable, etc. I also decided to ignore the discussion of what an integral is - but know that the derivative and the integral are related via the fundamental theorem of calculus and that an anti-derivative is the same as the indefinite integral. That's beyond the scope of this already long comment.)

Now, what's so special about the function y = e^x ? It turns out that if you get the derivative of y = e^x, you get back y = e^x . This is the only function in existence that does that.

So then there's the question of what the hell e is. You might already know that it's a mathematical constant the way pi is, but the significance of pi is kind of obvious - it's just the ratio of the diameter to the circumference of any circle. What the hell then is e?

e was discovered when a guy was thinking about continuous compounding interest. (Compounding interest is when the principle that you start with gets adjusted periodically so that the principle includes what you really started out with, and the interest that you've earned so far) If you have one dollar, at 100 percent interest, compounding once for 1 year, you get back 2 dollars at the end of the year, right?

Well, if you have it compounding quarterly, you get 2.44 dollars back. If you have it compound every month, you get 2.61 dollars back.

The dude that was doing calculations discovered that, it won't matter how many times you compound within a certain time frame, eventually you just don't get more money back. If your interest compounds every second in one year, you'll still get back only 2.72 (rounded) dollars.

As it turns out, if you put the compounding rate at infinite, (in other words, there are an infinite number of compounding times in one year) that final number approaches a constant of 2.718281828459..., which is e. That's where e comes from. (I forgot to mention: e is like pi, in that it never ends. When I first wrote the comment I accidentally gave the impression that it was a number with a terminating expansion)

e is special - it gets used in the base of so called 'natural logarithms'. Okay, now I'm getting lazy and telling you to look it up. I'm also getting honest and telling you that I don't know exactly why the derivative of e^x is equal to e^x , because I'm not that great at math. But I'm sure it has to do with the way e was defined.

There. That's the best I can do.

(Whoopsie poopsie, I forgot to explain that e is transcendental and has an infinite expansion)