Comment #12 on issue 1816 by [email protected]: Adding partial
derivatives and taking derivatives with respect to functions
http://code.google.com/p/sympy/issues/detail?id=1816
The simplest way of thinking about derivatives wrt to function is that
functions like f(x) and g(x) are *no different* that regular symbols. That
is f(x) => f and g(x) => g. Mathematically, this is what the chain rule
says and algorithmically, this is how the funcderiv branch implements these
derivatives. So the expression:
Derivative(f(g(x)), g(x))
is just:
df/dg * dg/dx
which is the chain rule. Your second example is just df/dg. In terms of
the diff(f(x), x).diff(f(x)) example, here is one way to think about it:
d/df df/dx = d/dx df/df = d/dx 1 = 0
The other way to think about it is that df/dx is, itself just a function
and thus in the context of a derivative can be treated just like a symbol,
call it u = df/dx. Then:
d/df u = du/df = 0
These types of manipulations forms the basis of the Lagrangian and
Hamiltonian approaches to mechanics as well as all of statistical mechanics
and thermodynamics. I don't want to minimize your questions, but there is
simply nothing dubious or ambiguous about these types of derivatives. If I
had a dollar for every time a physics student performs one of these
derivatives, I would be a rich man...
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