variable of differentiation returns True from `has_free` about sympy

[oscarbenjamin]

I would have expected both to be True.

[smichr]

While the integration symbol is truly a dummy symbol during definite integration, the symbol of differentiation is not. One must introspect to see if the symbol will survive differentiation. When the function being differentiated does not have the symbol of differentiation, t, the result clearly will be independent of t. That's why it makes sense that there is no t in the free symbols.

But if the function had t in a linear form then perhaps it is debatable whether the t should be reported as a free symbol or not, just like any other object, e.g. y=x**2 +2*x - (x + 1)**2 as written contains x, but after expansion, will not.

So the answer could be True or False for the OP expressions, but they should be consistent, and documentation should be clear for Derivative to indicate when a symbol will be considered free or not.

[asmeurer]

I don't think free symbols should try to do any sort of mathematical constant detection. It should be a fast structural check only, along the lines of ==. If the symbol appears in an expression in a place where it is free it should be considered a free variable.

Also, consider that if t is not free in expr, then that means that you should be able to replace any free symbol in expr with t and get an equivalent expression. But Derivative(f(x), t).subs(x, t) is not the same thing anymore. So from that point of view, even mathematically constant expressions should not be considered to be free of a given symbol if they still contain it literally somewhere.