**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 >>> f and the diagram on the bottom does f >>> 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)

**Proof of the chain rule.**

** 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 times 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(z ^{2} + 5z + 7) = 3(z^{2} + 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 x ^{2} – 4x + 3**

**Then the output is (x + 2)**

^{2}– 4(x + 2)x + 3**which is x**

^{2}– 1 (surprise, surprise)