Comment #11 on issue 2476 by asmeurer: nth order Derivative
http://code.google.com/p/sympy/issues/detail?id=2476
Yes, we should break compatibility on the internal representation.
Hopefully not too much outside of the sympy codebase uses it the way it is
now.
I think we should just extend fdiff() like you suggest.
By the way, doing this will not only make symbolic order differentiation
possible, but also very large numerical order differentiation. For
example, Derivative(f(x), x, 10000) has a very inefficient internal
storage. And cos(x).diff(x, 100000) is calculated very inefficiently.
Implementing a cos.fdiff(var, order) (btw, let's just call the
argument "order") to return cos, sin, -cos, or -sin depending on the
modulus of order with 4 would make it much more efficient. So we should
not only implement rules for composition, products, powers, etc., but also
optimize fdiff(var, order) where ever possible in existing functions.
By the way, can the chain rule be generalized to the nth order? I know the
product rule can be with the Leibniz rule, but I'm not sure off the top of
my head what diff(f(g(x)), (x, n)) would be in the general case.
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