Performing Automatic Differentiation by way of Simple Example

Question

I understand the theory of symbolic and numerical differentiation pretty well, however am struggling on understanding automatic differentiation.

I have read the following example in documentation and struggle with how certain steps are performed and require clarity.

By way of example, say we want to find the solution (using automatic differentiation) of $f'(3)$ where $f(x)$ is equal to:

$$f(x) = \frac{(x+1)(x-2)}{x+3}$$

My understanding is to use automatic differentiation method, prior to solving for $f'(x)$, we must understand two things:

1) We must work with the concept of "value pairs" when evaluating expressions i.e. $$\vec{u} = (u, u')$$ where u denotes the value of the function at a point $x_{0}$

2) For the purposes of this simple example, understand the two symbolic differentiation rules: $$\vec{x} = (x, 1)$$ $$\vec{c} = (c, 0)$$

Now using the rules above we evaluate as follow's: Example

In that example, how did they get the value of 5 in the expression: $\frac{(4, 5)}{(6, 1)}$ ? and how did they get a result of 13/18?

Answer 1

You need to employ rules for addition, multiplication, etc. as obtained from the basic rules for differentiation, that is $$\begin{align} (a,a')+(b,b')&=(a+b,a'+b')\\ (a,a')-(b,b')&=(a-b,a'-b')\\ (a,a')\cdot (b,b')&=(ab,a'b+ab')\\ \frac{(a,a')}{(b,b')}&=\left(\frac ab,\frac{a'b-ab'}{b^2}\right)\\ \end{align} $$ Thus $$(4,1)\cdot (1,1)=(4\cdot 1,1\cdot 1+4\cdot 1)=(4,5). $$

Answer 2

The differentiation rule for a quotient is

$$\left(\frac pq\right)'=\frac{p'q-pq'}{p^2},$$

which can be written

$$\frac{(p,q)}{(r,s)}\to\left(\frac pr,\frac{qr-ps}{r^2}\right).$$