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How do CAS evaluate derivatives

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matqkks
#1
Feb18-12, 01:25 PM
P: 153
How do CAS systems and programmable calculators evaluate the derivative of a function?
Do they use matrix representation of linear transformations?
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Stephen Tashi
#2
Feb19-12, 11:46 AM
Sci Advisor
P: 3,282
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
#3
Feb19-12, 03:01 PM
P: 153
Sorry for being vague but I meant symbolically.

joeblow
#4
Feb19-12, 06:04 PM
P: 71
How do CAS evaluate derivatives

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
#5
Feb19-12, 10:26 PM
Sci Advisor
P: 827
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
[itex]\sin(x^2)+3[/itex]
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]


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