Simplified Helmholtz decomposition

Question

Suppose we are given a vector field $$\vec{a}= (x+y+z)\vec{i}+y\vec{j}+z\vec{k} \tag{$*$}$$ And we want to write it as a linear combination of a potential field and a solenoidal field, meaning $$\vec{a}= \vec{p}+\vec{s}$$ where $$\vec{p}= \nabla\phi $$ with $\phi$ being the potential of the corresponding potential field. I first applied the divergence operator to both sides of eqn $\left (* \right )$ and got $$\operatorname{div}\vec{a}= \operatorname{div}(\operatorname{grad}(\phi))=3$$ Now my goal is to find the potential function. I've read in a paper, that to do so, you just have to perform antiderivation twice on $\operatorname{div} (\operatorname{grad}(b))$. Doing so I got that $$\phi = \frac{3}{2}x^2+C$$ However, what's bugging me is the fact that $\operatorname{div} (\operatorname{grad}(\phi))= \Delta b$ which is the Laplacian, and it involves not only the partial derivatives with respect to $x$. What happened here to the other two variables? Is this approach valid?