Steps Feynman omits in his "Tips on Physics"

Let me start with my exposition of a basic result about differentiation:

Suppose $u$ and $v$ are two functions from $\R$ to $\R$: $u,v : \R \to \R$.

Then we can form their (pointwise) product, and then differentiate that.

The product rule for differentiation, followed by an elementary algebraic manipulation, then yields:

$$(uv)' \xlongequal {\text{product rule}} u'v + uv' \xlongequal{\text{algebra}} uv(u'/u+v'/v)$$

That combined the product rule with some simple algebraic manipulations.

That is a simple, general, fact which may or may not be of general significance.

Now let's see what Feynman does in (1.3) through (1.6) of "Feynman's Tips on Physics".

He both generalizes and specializes the above.

He generalizes it to a four-way product:

$$(uvwx)' = u'vwx + uv'wx + uvw'x + uvwx' = uvwx \Big[ {u' \over u} + {v' \over v} + {w' \over w} + {x' \over x} \Big]$$

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Next he specializes to the casa where each of $u,v,w,x$ are powers of a polynomial, e.g. $u = p^a$, where $p$ is a polynomial and $a$ is a fixed, constant real number.

To differentiate $p^a$, we need to realize that that is the composite of two functions, the polynomial $p$ and the monomial $x^a$.

Let us write the latter as $x^a = xR_a$ ($x$ raised to the $a$-th power).

We differentiate that function as follows:

$$(x^a)' = x{R_a}' = x a {R_{a-1}} = a x^{a-1} .$$

Now we can apply the chain rule to differentiate the composite:

$$(p^a)' = (pR_a)' \xlongequal {\text{chain}} p' \cdot (p{R_a}') = p' \cdot (a(pR_{a-1})) = p' \cdot (a p^{a-1}) .$$

And so, if $u = p^a$, we have 

$${u' \over u} = {(p^a)' \over p^a} = { p' a p^{a-1} \over p^a } = a {p' \over p} .$$

And this is what Feynman applies four times to get his (1.6).

Juxtaposition, multiplication, application, and composition

Juxtaposition is certainly a common, efficient way of combining two things.

I want to give here three separate examples of such.

1. Multiplication 

If $a$ and $b$ are two real numbers, we routinely write $ab$ to denote their (multiplicative) product.

If necessary, we put a centered dot between them to prevent ambiguity.

Thus, for example, $5\cdot 7 = 35$, while $57$ is just that.

2. Function application.

Suppose $x$ is a real number, and $f:\R\to\R$ is a real-valued function of one real variable.

It is convenient to write $xf$ for the value of $f$ at $x$.

Of course more conventional ways of writing this are $(x)f$ or $f(x)$, depending on whether you are letting functions operate to the right or left of their argument.

3. Composition 

Suppose we have three sets $X, Y$ and $Z$, 

and functions $f:X\to Y$ and $g:Y\to Z$.

We can then define a composite function $fg$ by $x(fg) = (xf)g$.

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. $x(fg)' = (xf')(xg) + (xf)(xg')$.