Yes, I know that the proper way to do this is to prove the derivative formula for and then use, say, the implicit function theorem or perhaps the chain rule.

But an early question asked students to use the difference quotient method to find the derivative function (ok, the “gradient”) for And yes, one way to do this is to simplify the difference quotient by factoring from both the numerator and the denominator of the difference quotient. But this is rather ad-hoc, I think.

So what would one do with, say, where are positive integers?

One way: look at the difference quotient: and do the following (before attempting a limit, of course): let at which our difference quotient becomes:

Now it is clear that is a common factor..but HOW it factors is essential.

So let’s look at a little bit of elementary algebra: one can show:

(hint: very much like the geometric sum proof).

Using this:

Now as

we have (for the purposes of substitution) so we end up with:

(the number of terms is easy to count).

Now back substitute to obtain which, of course, is the familiar formula.

Note that this algebraic identity could have been used for the old case to begin with.