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It's true that the global function diff applies only to symbolic expressions, not to coordinate functions as defined in SageManifolds. But you can use the method diff on them:

sage: Lx[3].diff(th)
-sin(ph)/sin(th)^2

Note that the result is still a coordinate function:

sage: type(Lx[3].diff(th))
<class 'sage.manifolds.coord_func_symb.CoordFunctionSymb'>

To get a symbolic expression, use the method expr():

sage: Lx[3].diff(th).expr()
-sin(ph)/sin(th)^2
sage: type(Lx[3].diff(th).expr())
<type 'sage.symbolic.expression.Expression'>

Another solution is to invoke expr() prior to the global function diff:

sage: diff(Lx[3].expr(), th)
-cos(th)^2*sin(ph)/sin(th)^2 - sin(ph)

Note that in this case, the result is not automatically simplified (this is one of the differences between symbolic expressions and coordinate functions).