I have some expressions with grad and div operators and would like to re-write the expressions so that there are no grads or divs. Basically, I have been told that $\nabla \phi$, where $\phi = 1/4 \pi |\textbf{r}|$, can be re-written as
$\nabla \phi = \frac{\textbf{r}}{|\textbf{r}|} \frac{\text{d}\phi}{\text{d}|\textbf{r}|}$.
I might need to refresh my knowledge of vector calculus, but I suppose this just follows from the chain rule?
Edit:
$\nabla \nabla \phi = \frac{\nabla (\textbf{r})}{|\textbf{r}|}\frac{\partial}{\partial r}\phi + \hat{r} \nabla \bigg( \frac{\partial}{\partial r} \phi \bigg)=\frac{\textbf{r}}{|\textbf{r}|} \frac{\text{d} \phi}{\text{d} |\textbf{r}|} + \frac{\textbf{r}}{|\textbf{r}|}\frac{\textbf{r}}{|\textbf{r}|} \frac{\partial}{\partial r} \bigg( \frac{\partial}{\partial} \phi \bigg) = \frac{\delta_{ij}}{|\textbf{r}|}\frac{\text{d} \phi}{\text{d}|\textbf{r}|} + \frac{\textbf{r} \otimes \textbf{r}}{|\textbf{r}|^2} \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} , $
$\nabla \nabla \nabla \phi = \nabla \bigg( \frac{\delta_{ij}}{|\textbf{r}|}\frac{\text{d} \phi}{\text{d}|\textbf{r}|} + \frac{\textbf{r} \otimes \textbf{r}}{|\textbf{r}|^2} \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} \bigg)=\nabla \bigg(\frac{\delta_{ij}}{|\textbf{r}|} \bigg) \frac{\text{d}\phi}{\text{d} |\textbf{r}|} + \frac{\delta_{ij}}{|\textbf{r}|} \nabla \bigg(\frac{\partial \phi}{\partial r} \bigg) + \nabla \bigg(\frac{\textbf{r} \otimes \textbf{r}}{|\textbf{r}|^2} \bigg) \frac{\partial^2 \phi}{\partial r^2} +\frac{\textbf{r} \otimes \textbf{r}}{|\textbf{r}|^2} \nabla \bigg( \frac{\partial^2 \phi}{\partial r^2} \bigg) = \nabla \bigg(\frac{\delta_{ij}}{|\textbf{r}|} \bigg) \frac{\text{d}\phi}{\text{d} |\textbf{r}|} + \frac{\delta_{ij} \otimes \textbf{r}}{|\textbf{r}|^2} \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} + \frac{\delta_{ij} \otimes \textbf{r}}{|\textbf{r}|^2} \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} + \frac{\textbf{r} \otimes \delta_{ij} }{|\textbf{r}|^2} \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} + \frac{\textbf{r} \otimes \textbf{r} \otimes \textbf{r}}{|\textbf{r}|^3} \frac{\text{d}^3 \phi}{\text{d} |\textbf{r}|^3} , $
$\nabla \cdot \nabla \phi = \nabla^2 \phi = \frac{\text{d}^2 \phi}{\text{d} |\textbf{r}|^2} $.
Your expression for the gradient will work with any function that is a function purely of the radius (i.e., no angular dependencies). In spherical coordinates:
$\displaystyle \nabla\Phi = \hat{r}\frac{\partial}{\partial r}\Phi + \hat{\theta}\frac{\partial}{\partial \theta}\Phi+\hat{\phi}\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\Phi $
If $\Phi$ is a function of $r$ only,
$\displaystyle \nabla\Phi = \hat{r}\frac{\partial}{\partial r}\Phi$
But $\hat{r}=\bf{r}/|\bf{r}|$ and $\partial/\partial r = d/d|\bf{r}|$. Make those substitutions and you get your expression.