$$ \frac{\partial \varphi }{\partial r} = \frac{\partial}{\partial r}\left.\left(\frac{q}{\sqrt{r^{2}+b^{2}-2rb\cdot\cos(\theta)}}-\frac{q}{\sqrt{r^{2}+b^{2}+2rb\cdot\cos(\theta)}}+\right.\right. $$ $$ \left.\left.\frac{-q\frac{a}{b}}{\sqrt{r^{2}+(\frac{a^{2}}{b})^{2}-2r\frac{a^{2}}{b}\cdot\cos(\theta)}}-\frac{-q\frac{a}{b}}{\sqrt{r^{2}+(\frac{a^{2}}{b})^{2}+2r\frac{a^{2}}{b}\cdot\cos(\theta)}}\right)\right|_{r=a}= $$ $$ \left.q\left(\frac{r-b\cdot\cos(\theta)}{\left(r^{2}+b^{2}-2rb\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{r+b\cdot\cos(\theta)}{\left(r^{2}+b^{2}+2rb\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\right.\right. $$ $$ \left.\left.\frac{\frac{a}{b}\left(r-\frac{a^{2}}{b}\cdot\cos(\theta)\right)}{\left(r^{2}+(\frac{a^{2}}{b})^{2}-2r\frac{a^{2}}{b}\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\frac{a}{b}\left(r+\frac{a^{2}}{b}\cdot\cos(\theta)\right)}{\left(r^{2}+(\frac{a^{2}}{b})^{2}+2r\frac{a^{2}}{b}\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)\right|_{r=a}= $$ $$ q\left(\frac{a-b\cdot\cos(\theta)}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{a+b\cdot\cos(\theta)}{\left(a^{2}+b^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}- \right. $$ $$ \left.\frac{\left(\frac{a}{b}\right)^{2}\left(b-a\cdot\cos(\theta)\right)}{\left(\frac{a}{b}\right)^{3}\left(b^{2}+a^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\left(\frac{a}{b}\right)^{2}\left(b+a\cdot\cos(\theta)\right)}{\left(\frac{a}{b}\right)^{3}\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)= $$ $$ q\left(\frac{a-b\cdot\cos(\theta)}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{\frac{b}{a}\left(b-a\cdot\cos(\theta)\right)}{\left(b^{2}+a^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\right. $$ $$ \left.\frac{a+b\cdot\cos(\theta)}{\left(a^{2}+b^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{\frac{b}{a}\left(b+a\cdot\cos(\theta)\right)}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)= $$ $$ q\left(\frac{a^{2}-b^{2}}{a\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}+\frac{b^{2}-a^{2}}{a\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)= $$ $$ q\frac{b^{2}-a^{2}}{a}\left(\frac{1}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{1}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right). $$