If I have V(z,t) + V(z+ delta(z),t) and I take the limit as z goes to zero, do I have 2V(z,t) or dV(z,t)?

Also, if I have dV(z,t)/dt + dV(z+delta(z),t)/dt and the limit z is going to 0, do I have 2dV(z,t)/dt or d^2V(z,t)/dt^2 ?