Algebra with differential operators (Alternative forms of the Laplacian in spherical coordinates) Given is the following: $$\frac{1}{r^2} 
\frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right) =  
\frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial r} = \frac{1}{r} 
\frac{\partial^2}{\partial r^2} \left( r  \, f \right)$$
My question is: How can I get to the term on the right $\frac{1}{r} 
\frac{\partial^2}{\partial r^2} \left( r  \, f \right)$ from the left term $\frac{1}{r^2} 
\frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right)$ or the term in the middle $\frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial r}$?
Using the product rule I can easily get from the left term $\frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right)$ to the term in the middle $\frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial r}$. And I also can get from the term on the right $\frac{1}{r} 
\frac{\partial^2}{\partial r^2} \left( r  \, f \right)$ to the term in the middle. So I can verify that the given relation is true. But how to get the the term on the right in the first place is beyond my knowledge.
Background:
The Laplacian in spherical coordinates $( r ,\vartheta ,\phi )$ is
$$\Delta f = \frac{1}{r^2} 
\frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right) +
\frac{1}{r^2 \sin \vartheta}  \frac{\partial}{\partial \vartheta} \left(\sin\vartheta \, \frac{\partial f}{\partial \vartheta} \right) +
\frac{1}{r^2 \sin^2\vartheta}  \frac{\partial^2 f}{\partial \phi^2}\,.
$$
Some problems are easier to solve when the first term is in one of the alternative forms given above. However I don't want to memorize all three forms. I want to remember just the first form, and if I need the others, just derive them with a little bit of algebra.
 A: It can be demonstrated that:
\begin{align}\tag{1}
\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right) &= \frac{1}{r^{2}} \, \left( r^{2} \, \frac{\partial^{2} f}{\partial r^{2}} + 2 \, r \, \frac{\partial f}{\partial r} \right) \\
&= \frac{\partial^{2} f}{ \partial r^{2}} + \frac{2}{r} \, \frac{\partial f}{\partial r}
\end{align}
and
\begin{align}
\frac{1}{r} \frac{\partial^2}{\partial r^2} \left( r  \, f \right) &= \frac{1}{r} \, \frac{\partial}{\partial r} \left( r \, \frac{\partial f}{\partial r} + f \right) \tag{2} \\ 
&= \frac{1}{r} \left( r \frac{\partial^{2} f}{\partial r^{2}} + 2 \, \frac{\partial f}{\partial r} \right) \\
&= \frac{\partial^{2} f}{ \partial r^{2}} + \frac{2}{r} \, \frac{\partial f}{\partial r}. \tag{3}
\end{align}
By rearranging the terms, ie starting with (1) and writing in reverse order (3) to (2), one state
\begin{align}
\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2  \,\frac{\partial f}{\partial r} \right) = \frac{1}{r} \frac{\partial^2}{\partial r^2} \left( r  \, f \right)
\end{align}
