How small must Dx be for Df/Dx to be df/dx?

There are two answers: 

1. If f(x) lies on the tangent line to f at x₀ , to the accuracy of your computations, then Dx is good enough to give the correct slope.  

2. You may always choose Dx to be an "infinitesimal", that is a number much smaller than any number associated with your problem, and so small that its square is indistinguishable from 0 to your criterion of accuracy.