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How do CAS evaluate derivatives

  1. Feb 18, 2012 #1
    How do CAS systems and programmable calculators evaluate the derivative of a function?
    Do they use matrix representation of linear transformations?
     
  2. jcsd
  3. Feb 19, 2012 #2

    Stephen Tashi

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    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?
     
  4. Feb 19, 2012 #3
    Sorry for being vague but I meant symbolically.
     
  5. Feb 19, 2012 #4
    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.
     
  6. Feb 19, 2012 #5

    pwsnafu

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    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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