Divergence of the gradient of f

Question

I just learned about the Laplacian, curl and divergence operators and frowned confusingly after stumbling on this notation: $ \bigtriangleup f(x) = ( (\nabla \cdot \nabla )f )(x) = \nabla \cdot (\nabla f) (x)$. The thing is, when I try to expand the second term I get: $( (\nabla \cdot \nabla )f )(x) = ((\partial^{2}_{xx} + \partial^{2}_{yy} + \partial^{2}_{zz}) f ) (x)$. The RHS clearly shows we're dealing with a scalar and if we multiply $ f $ with the operators we'd end up with the definition of the Laplacian, which is fine. Now, I'm really wondering if this$(\partial^{2}_{xx} + \partial^{2}_{yy} + \partial^{2}_{zz})$ is legit. Maybe it's just physics notation, which isn't super pedantic and rigourous, or maybe I just got it all wrong. I'd appreciate it if soemone could explain where I messed up. Thanks !