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