The original motivation for derivatives wrt functions and derivatives of functions was to support the Euler-Lagrange equations https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation. In Euler-Lagrange, dL(t, q(t), q'(t))/dq'(t) means consider L as a function of three variables, L(x, y, z) and take dL/dz evaluated at z=q'(t).
See also https://github.com/sympy/sympy/issues/15048 Aaron Meurer On Wed, Aug 29, 2018 at 11:52 AM, Robert Dougherty-Bliss <[email protected]> wrote: > Why do we assume that g'(t).diff(g(t)) == 0? > > Here's a question on math.SE about derivatives w.r.t. functions: > https://math.stackexchange.com/questions/954073 > > Does the assumption work with the accepted answer there? > > > On Wednesday, August 29, 2018 at 2:12:07 AM UTC-4, Aaron Meurer wrote: >> >> I believe they are equal, according to SymPy's rule that >> (g'(t)).diff(g(t)) == 0: >> >> f(g(t)).diff(t, g(t)) == (f'(g(t))*g'(t)).diff(g(t)) == f''(g(t))*g'(t) >> >> f(g(t)).diff(g(t), t) == f'(g(t)).diff(t) == f''(g(t))*g(t) >> >> You can also verify this with SymPy: >> >> >>> f(g(t)).diff(t).diff(g(t)) >> Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t) >> >>> f(g(t)).diff(g(t)).diff(t) >> Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t) >> >> Aaron Meurer >> >> >> On Tue, Aug 28, 2018 at 11:55 PM, Chris Smith <[email protected]> wrote: >> > SymPy allows derivative wrt non-Symbols. Under the current assumptions, >> > `g(t).diff(g(t),t) == g(t).diff(t, g(t)) == 0`. Can anyone give an >> > example >> > where `f(g(t)).diff(t, g(t))` would not equal `f(g(t)).diff(g(t), t)`? >> > >> > -- >> > You received this message because you are subscribed to the Google >> > Groups >> > "sympy" group. >> > To unsubscribe from this group and stop receiving emails from it, send >> > an >> > email to [email protected]. >> > To post to this group, send email to [email protected]. >> > Visit this group at https://groups.google.com/group/sympy. >> > To view this discussion on the web visit >> > >> > https://groups.google.com/d/msgid/sympy/73c688ce-d447-4767-8ded-373346b822b7%40googlegroups.com. >> > For more options, visit https://groups.google.com/d/optout. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/22f5c508-94ba-44f9-bd73-0d1f65f90330%40googlegroups.com. > > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CAKgW%3D6LQqn%2BV7Jda27SmLBK614jsVW0v3cirvQTx4CNHTGZT_A%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
