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).