```Christine

>(2) f(x+h) = f(x) + K(x,h)h
>K(x,h) = f'(x) + Q'(x,h)  <-- How this is derived from step 2 above.  I
>don't understand how it becomes the derivative of f and Q.

It doesn't "derive". There is (or should have been) a big "IF". (We don't
have a good notation. Just as := means "is defined to be", we need something
like *= for "expands to").

So, this says that IF K(x,h) can be put into this form. The form on the
right hand side is visibly
"a function just of x named f'" plus "a null function named Q'".

Thus, this line is equivalent to the limit expression

lim K(x,h) = f'(x)
h->0

Now, look what you get if you substitute the expansion of K into
the expansion of f(x+h), subtract f(x) from both sides, divide by h
(provided that makes sense), and express it as a limit, you get

f(x+h) - f(x)
lim  -------------   =  f'(x)
h->0       h

which you recognize from baby calculus.

The THEN for this IF is a definition of that which you called f'(x) is
the derivative of f. (The traditionalists call it the "total" derivative
because is components (remember, in general it's a matrix), are the
"partial" derivatives.)

This exposition makes is look like your are calculating the derivative
rather than pulling it out of a hat, or evaluating the limit of a
fraction whose numerator and denominator both vanish. It buys you a
way of doing the multivariate differential calculus with a minimum
of "re-tooling" your notation. Instead of a new notation, you get a
new interpretation. (If you you are familiar with the lingo of
object oriented programming, this is a perfect case of "overloading"
the differentiation operation.)

OK?

G.Francis
```