Hi,
For linear programming (LP) a variety of techniques may be used, including the simplex and interior point methods. Gurobi can use both as far as I am aware (see here). As for solving LPs, derivatives are not needed when using the simplex method. Essentially the method moves from vertex to vertex in the feasible space until it cannot improve the objective anymore. This is a mostly algebraic procedure, e.g. see here.
However, finding gradients of linear or even quadratic functions is really simple so if they are required they can likely be calculated automatically exactly without much fuss. I hope this clears it up!