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In differential calculus, the product rule is both simple in form and high in utility. As such, it is typically presented early on in calculus courses — soon after the linearity of the derivative, in fact. Moreover, the product rule is easy to derive from first principles:

**Theorem (Product Rule): ***Let and be differentiable on the open set . Then is differentiable on , and we have*

for all *.*

*Proof: *For , we have (by definition of the derivative)

under the assumption that each of these last two limits exists. This of course holds, as these limits are and , respectively.

All in all, then, the product rule is easy to prove and easy to use. But — and this is of utmost pedagogical importance — * is the product rule intuitive? *By this proof alone, I would argue not; the manipulation of the numerator is weakly-motivated and our result falls out without reference to more general phenomena.

In this post, we’ll explore the merits of a second proof of the product rule, one that I hope presents a motivated and compelling argument as to **why**** **the product rule should look the way it does.