Hint: This simplest way to prove eq. (1) is to use the formulas $$[a(x),N]~=~a(x)\qquad\text{and}\qquad [N, a^{\dagger}(x)]~=~a^{\dagger}(x)$$ directly in eq. (2), and that the number operator $N$ commutes with the derivative $\nabla_x^2$.

Hint: This simplest way to prove eq. (1) is to use the formulas $$[a(x),N]~=~a(x)\qquad\text{and}\qquad [N, a^{\dagger}(x)]~=~a^{\dagger}(x)$$ directly in eq. (2), and that the number operator $N$ commutes with the derivative $\nabla_x^2$.

(In this way we avoid dealing with derivatives of the Dirac delta distribution. We should stress that the latter is also a feasible route with appropriate care.)