Saturday, January 26, 2013

The derivative of products

An elementary proof 

Knowing some important formulas by heart can be very useful, but if one knows how to derive them, it is no longer necessary to remember the formula. From reading math textbooks (many  or most of them?), one can gain the false impression that the process of deriving a formula follows the same sequence as the proof.

Here is an elementary proof that nevertheless involves some not so obvious steps. 

Suppose that

then the formula for the derivative is 

but how do we prove this? Although the proof is straightforward, it is perhaps difficult to remember all the tricks that are required and when to apply them. Here is a standard proof. First, apply the definition of derivative to the product of the two functions:

The next step is the crucial operation, at once trivial and far from obvious. We are going to both subtract and add u(x)v(x+h) and rewrite the ratio as

Now, who would think of adding two terms that sum to 0 into such an expression? This is an idea that doesn't make much sense at this point. Indeed, one needs to look a few steps ahead and see what it is going to be needed for. What follows are just some simple factorizations of terms.

Break out some terms to get 

then take limits and replace derivatives, and we are done:

Here, the simple formula seems much easier to memorize than all the steps of the proof. (In fact, you may impress your friends far more if you memorize Hugo Ball's poem Karawane than if you learn to recite the steps of this proof.)

It is highly misleading when formulas such as the above are just plainly stated and then concisely proven. This is most likely not how the formulas were originally discovered. Rather, one would observe a few instances of derivatives of multiplied functions and conjecture a formula. Then, starting from the formula as well as the definition of derivative, one would work backwards and find all the arithmetic manipulations that make the proof work.

Instead of learning a fixed set of steps that are used in particular proofs, one would probably learn a bag of tricks that can be applied in various situations. Then, out of this bag one can grab various operations that can be tried out, until something is found that leads the proof in a promising direction.

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