Morse and Feshbach in their classic book, pp. 6$-$7, give a definition of the second derivative $d^{2}\psi/dx^{2}$ in one-dimension for a scalar field $\psi(x)$ as being related to the difference between the value of $\psi$ at $x$ and its average at neighbouring points:
$$\lim \{ \psi(x)-\frac{1}{2}[ \psi(x-dx)+\psi(x+dx) ]\} \\ = -\frac{1}{2}\lim \{[\psi(x+dx)-\psi(x)]-[\psi(x)-\psi(x-dx)]\}\\ =-\frac{1}{2}(d^{2}\psi/dx^{2})(dx)^{2}$$
This is clear enough and understandable.
But now, they try to generalise the concept into three-dimensions to reach the definition of Laplacian operator, but I have difficulty reaching the same result. They say that in three dimensions, the equivalent result will be:
$$-\frac{1}{6} (dx\ dy\ dz)^{2} \left(\frac{d^{2}\psi}{dx^{2}}+\frac{d^{2}\psi}{dy^{2}}+\frac{d^{2}\psi}{dz^{2}}\right)=-\frac{1}{6} (dx\ dy\ dz)^{2} \nabla^{2}\psi$$
I am not sure how this was derived, because if we follow similar lines to the initial one-dimensional derivation above and use neighbouring points like $\psi(x\pm dx,y,z),\psi(x,y\pm dy,z)$ and $\psi(x,y,z\pm dz)$, we end up with: $-\frac{1}{6} \left(\frac{d^{2}\psi}{dx^{2}}(dx)^{2}+\frac{d^{2}\psi}{dy^{2}}(dy)^{2}+\frac{d^{2}\psi}{dz^{2}}(dz)^{2}\right)$. I am not clear on how they managed to isolate the factor $(dx\ dy\ dz)^{2}$ in their solution.