Comment #39 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
We have already established that derivatives wrt to function does *not*
commute with algebraic manipulations. Take for example, the Hamiltonian:
H(p(t),x(t)) = p(t)**2/2*m + k*x(t)**2/2
It's derivatives wrt to x(t) and p(t) give the differential equations
obeyed by x(t) and p(t). But we know that H(p(t),x(t)) = E, which is a
constant. If you use the algebraic relationship before taking the
derivatives, you get 0 for all the derivatives and don't get the proper
differential equations.
There is nothing special about known functions like sin/cos in this
respect. You simply can't do algebraic manipulations and expect to get the
same derivatives wrt functions.
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