Definition of grad^2 : (grad)^2 f = fxx+fyy+fzz where the subscripts mean partial derivatives.
For example, fx=∂f/∂x.
Note that r2=x2+y2+z2, so taking partial derivatives of both sides ...
2rrx=2x => rx=x/r and likewise ry=y/r and rz=z/r
So, taking the first partial of f(r) wrt x ....
fx = frrx = (x/r) f'(r) where ∂f/∂r = fr = f'(r) since f only depends on r.
fxx = ∂/∂x [ (x/r) f'(r) ] = (1/r)f' + x ∂/∂x [ (1/r) f'(r) ] by Product Rule
= (1/r)f' + x [(1/r)f']rrx by Chain Rule
= (1/r)f' + x [ (1/r)f'' - (1/r2)f' ] (x/r) again by Product rule on the ∂/∂r
⇒ fxx = (1/r -x2/r3)f' + (x2/r2)f''
and fyy = (1/r -y2/r3)f' + (y2/r2)f''
and fzz = (1/r -z2/r3)f' + (z2/r2)f''
adding these yields and terms we will see only depend on r:
So fxx+fyy+fzz = (3/r - (x2+y2+z2)/r3)f' + ((x2+y2+z2)/r2)f''
= (3/r - 1/r) f' + f'' since x2+y2+z2 = r2
= (2/r)f' + f''
= (2/r) ∂f/∂r + ∂2f/∂r2 Your result, written in more standard form.