Draft

DRAFT, just so I can see how the mathjax is processed.

Richard Phillips Feynman has a near revered reputation in science pedagogy, for very good reasons, which I wholeheartedly endorse.

But he did, in at least one case, miss the key points and issues.

The two basic, necessary, facts:

The product rule 

and the chain rule.

(Here we only consider real-valued functions of one real variable, that is functions $f: \R \to \R$.)

The product rule is this:

Suppose we have two functions $f,g : \R \to \R$, and form their product $x(fg) = (xf)(xg)$.

Here the key point is that we are using juxtaposition to denote both function application $xf = (x)f$, i.e. the function $f$ applied for the variable $x$, 

and multiplication of real numbers, i.e., $(xf)(xg)$ is the real number $xf$ times, i.e., multiplied by the real number $xg$.

Now the product rule answers the question, what is $(fg)'$.

Answer

$(fg)' = f'g + fg'$, i.e. $x(fg)' = (xf')(xg) + (xf)(xg')$.