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The problem could be with how f was defined.
In particular if it was defined as an expression rather than a function.
Be it with SageMath 10.2 or SageMath 10.4, the following input
sage: b, c = SR.var('b, c')
sage: f = function('f')
sage: equation = (b - f(0))*(c - f(0))*D[0](f)(0) - 1 == 0
sage: new_equation = equation.subs({f(0): b + c})
sage: solve(new_equation, D[0](f)(0))
gives the following output
[D[0](f)(0) == 1/(b*c)]
