Definition of grad^2 : (grad)^2 f = f_{xx}+f_{yy}+f_{zz} where the subscripts mean partial derivatives.

For example, f_{x}=∂f/∂x.

Note that r^{2}=x^{2}+y^{2}+z^{2}, so taking partial derivatives of both sides ...

2rr_{x}=2x => r_{x}=x/r and likewise r_{y}=y/r and r_{z}=z/r

So, taking the first partial of f(r) wrt x ....

f_{x} = f_{r}r_{x} = (x/r) f'(r) where ∂f/∂r = f_{r} = f'(r) since f only depends on r.

f_{xx }= ∂/∂x [ (x/r) f'(r) ] =_{ }(1/r)f'_{ }+ x ∂/∂x [ (1/r) f'(r) ]_{ } by Product Rule

_{ }= (1/r)f' + x [(1/r)f']_{r}r_{x} by Chain Rule

= (1/r)f' + x [ (1/r)f'' - (1/r^{2})f' ] (x/r) again by Product rule on the ∂/∂r

⇒ f_{xx} = (1/r -x^{2}/r^{3})f' + (x^{2}/r^{2})f''

and f_{yy }=_{ }(1/r -y^{2}/r^{3})f' + (y^{2}/r^{2})f''

and f_{zz }=_{ }(1/r -z^{2}/r^{3})f' + (z^{2}/r^{2})f''

adding these yields and terms we will see only depend on r:

So f_{xx}+f_{yy}+f_{zz} = (3/r - (x^{2}+y^{2}+z^{2})/r^{3})f' + ((x^{2}+y^{2}+z^{2})/r^{2})f''

= (3/r - 1/r) f' + f'' since x^{2}+y^{2}+z^{2} = r^{2}

= (2/r)f' + f''

= (2/r) ∂f/∂r + ∂^{2}f/∂r^{2} Your result, written in more standard form.