How Do CAS and Programmable Calculators Evaluate Derivatives?

[matqkks]

How do CAS systems and programmable calculators evaluate the derivative of a function?
Do they use matrix representation of linear transformations?

 

[Stephen Tashi]

I don't know the answer, but you should specify whether you are asking how they evaluate derivatives numerically or how they evaluate them symbollically. Is the result of the evaluation a formula? Or a graph? Or a numerical table?

 

[matqkks]

Sorry for being vague but I meant symbolically.

 

[joeblow]

I suspect that they convert whatever expression you want to differentiate into taylor series, differentiate (in the obvious way), then match the result to a taylor series that represents an elementary function and substitute back. Maybe not, but I can't imagine how else it would be done.

 

[pwsnafu]

From what I heard CAS stores the information as a directed graph. In Mathematica you can use the FullForm command to see it directly for example
$\sin(x^2)+3$
would be
Plus[3,Sin[Power[x,2]]]
It then has rules for how to manipulate these objects. So the derivative operator D (I'm assuming wrt x) interacts with Plus via the rule
D[Plus[f,g]] = Plus[D[f],D[g]]
Mathematica knows that 3 is constant and so D[3]=0. It then reduces Plus[0,?] to just ?.
So we now have
D[Sin[Power[x,2]]]
It allies its chain rule and is programmed so that D[Sin] = Cos:
Multiply[Cos[Power[x,2]],D[Power[x,2]]]
And we know that the derivative of Power[x,2] as Multiply[2,x]