A function $\vec{F}$ is dependent on the lenght (norm) of $\vec{r}$
${\vec{F}(\vec{r})=F_{x}(\vec{r})\hat{x}+F_{y}(\vec{r})\hat{y}+F_{z}(\vec{r})\hat{z}} $,
in which
$F_{x}=F\left(\vec{r}\right)\frac{r_{x}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}$, $F_{y}=F\left(\vec{r}\right)\frac{r_{y}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}$, ${F_{z}=F\left(\vec{r}\right)\frac{r_{z}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}}$
Now I want to calculate the following derivative:
$\frac{\partial}{\partial r_x}\left(\vec{r}\cdot\vec{F}(\vec{r})\right)=\frac{\partial}{\partial r_x}\left(r_xF_x+r_yF_y+r_zF_z\right)=F_{x}+r_{x}\frac{\partial F_{x}}{\partial r_{x}}+r_{y}\frac{\partial F_{y}}{\partial r_{x}}+r_{z}\frac{\partial F_{z}}{\partial r_{x}} $
where $\frac{\partial F_{x}}{\partial r_{x}}=\frac{\partial F(\vec{r})}{\partial r_{x}}\frac{r_{x}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}+F(\vec{r})\frac{\partial}{\partial r_{x}}\left(\frac{r_{x}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}\right) $
in which $\frac{\partial}{\partial r_{x}}\left(\frac{r_{x}}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}\right) =\frac{1}{\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}}-\frac{r_x}{(r_{x}^{2}+r_{y}^{2}+r_{z}^{2})^{3/2}}$
When I use this in a model, the answers are incorrect, so it got me to think there is something wrong in my reasoning.