[crntaylor]

The typical approach is to provide overloaded versions of primitive functions (generally addition, multiplication, subtraction, division, powers, trig and hyperbolic trig functions, exponential and logarithm) for which you explicitly tell the program what the first N terms in the Taylor series are.

The magic is that you also tell it how to compute Taylor series of function compositions, if the Taylor series of the functions being composed are already known - then any arbitrary function composed out the primitive functions can have its Taylor series computed automatically!

For your example, the function 1/(1-x) is the composition of

    x -> -x
    x -> 1+x
    x -> 1/x

and so its Taylor series is already known as long as you have already defined negation, addition and reciprocation.

[jordigh]

So instead of implementing a full CAS that knows linearity of differentiation along with the product and chain rules, you implement something that for a little less computation can give you exact derivatives at any given point. Am I understanding this correctly?

[crntaylor]

Yes, that's more or less it. The system you implement still knows about linearity of differentiation, the product rule, chain rule etc but it's not a full blown CAS. It can't give you the symbolic derivative of a function (although I guess you can technically apply AD to any numeric type, including a symbolic type..) but it can compute the numerical value of the derivatives at arbitrary points.