# 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}

#### 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)