Be it with SageMath 10.2 or SageMath 10.4, the following input
b, c = SR.var('b, c')
f = function('f')
equation = (b - f(0))*(c - f(0))*D[0](f)(0) - 1 == 0
new_equation = equation.subs({f(0): b + c})
solve(new_equation, D[0](f)(0))
gives the following output
[D[0](f)(0) == 1/(b*c)]
What version of SageMath are you having trouble with?