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I'm wondering if an automatic differentiation package exists for Mathematica. This is what I mean by automatic differentiation.

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I don't know; but it's interesting to note that in Mathematica the distinction between a program and a symbolic expression is blurred. Many of the simple cases will just work with D/Derivative. It will be interesting to see some rudimentary implementations. See also forums.wolfram.com/mathgroup/archive/2008/Feb/msg00381.html –  Szabolcs Feb 16 '12 at 15:19
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To add to what Szabolcs said, for pure Mathematica expressions, symbolic differentiation is pretty much a generalization of automatic differentiation. I would guess that for these expressions, true automatic differentiation may not be any more efficient than using D. –  David Z Feb 16 '12 at 18:12
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In the link you provided are listed different packages, some of which are written in languages that can communicate with Mathematica. See demonstrations.wolfram.com/AutomaticDifferentiation and wolfram.com/products/applications/acegen –  CHM Feb 16 '12 at 23:41
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Try this. It may not be exactly what your looking for, but it also may give you a good starting point.

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