I am unable to resist saying a word or two about how we get the standard method. I can't emphasize enough that you don't need to know this to use calculus! But it might be interesting. Here goes...

Remember:

Where does this come from?

Well, what is the slope of the curve at any point x? Imagine that we increase x by a small amount, we'll call that amount h. The slope of the curve is the

*difference*in y [ the value of the function f(x) ], between the two points x and x + h. How much does it change?

Suppose:

Well,

So...

And there are three quick points:

• (1) The first term x

^{n}is what we need to subtract, it is f(x)

• (2) The third and following terms involve h

^{2}(and higher powers). We

*ignore these terms*!

The point is that h

^{2}goes away faster than h as h gets very small, and if you don't think it's getting small fast enough, just choose h to be even smaller...

• (3) What remains is the first term of the binomial expansion:

To get the slope, we need to divide by h:

The h / h part is still equal to 1 even when h gets very small. So it goes away!

There you have it.

As I said, this is for my son. Here is a picture from an earlier time: