The Chain Rule and the theory.

I found this account of the chain rule from Sam Shah

https://samjshah.com/2018/02/08/there-might-be-light-at-the-end-of-the-chain-rule-tunnel-maybe/

The partial text is as follows:

Find two points close to each other, like (x,g(f(x)) and (x+0.001,g(f(x+0.001)).
Find the slope between those two points: {g(f(x+0.001)-g(f(x))}/{(x+0.001)-x}.
There we go. An approximation for the derivative! (We can use limits to write the exact expression for the derivative if we want.)

But that doesn’t help us understand that {d/dx}[g(f(x)]=g'(f(x))f'(x) on any level. They seem disconnected!

But I’m on my way there. I’m following things in this way: x >>> f >>> g

Check out this thing I whipped up after school today. The diagram on top does x \rightarrow f and the diagram on the bottom does f \rightarrow g. The diagram on the right does both. It shows how two initial inputs (in this case, 3 and 3.001) change as they go through the functions f and g.
At the very bottom, you see the heart of this.

It has {δg}/{δf} times {δf}/{δx}={δg}/{δx}

(end of Sam Shah’s bit)
We can be more exact and use the derivatives, and show that the formula is true for y = g(f(x)) with f(x) = x ⁿ and g(f) = f ^m
The direct solution for y’ is the derivative of x ^mn, that is mnx ^{mn – 1}
The formula solution is mf ^{m – 1} times nx ^{n – 1}, giving mnx ^n(m-1) times x ^n-1
and finally mnx ^{mn – 1}

 

SOME DEFINITIONS:
A function is a process which converts an input into a corresponding output.
In symbols, input –> f –> output

Examples:
The input is transformed (converted) into the output,
usually by a formula or an expression using the values of the input:
output = 2(input) + 5
or you can write this as output = 2 x input + 5

The input and the output are both expressions, which can be
a: a number, or
b: a single variable, x or y or z … , or
c: a more complicated expression

The commonest form for the input-output relationship for a function f
is, as an example, f(x) = 3x + 4, where x is an input
and the corresponding output is 3x + 4

f is the label of the function,
and f(x) is the expression whose value is 3x + 4
f(x) = 3x + 4 is then an equation
The equation can be seen for example as f(8) = 3 x 8 + 4,
or f(y) =3y + 4 using y as the input,
or f(z² + 5z + 7) = 3(z² + 5z + 7) + 4 using an expression.

In its most simple formulation the input is not present and the equation is simply f = 3y +4, where the input
is identified as the ‘y’.

Now if an equation has a single variable on the left and an expression on the right then
a) it can be interpreted as a function (functional form) with f(y) = 3y + 4, and
b) the expression on the right can be substituted for the variable on the left.
Example
Let g(A) = A + 2 be a function g with output A + 2
Then it can be identified with the equation g = A + 2
Let g have the input x² – 4x + 3
Then the output is (x + 2)² – 4(x + 2)x + 3
which is x² – 1 (surprise, surprise)