Feynman lectures electric dipole question

[axmls]

For some reason, I'm having trouble with what I feel should be a relatively simple derivative to take. Feynman is differentiating the potential to find the z-component of the electric field. He has:
$$-\frac{\partial \phi}{\partial z} = - \frac{p}{4 \pi \epsilon_0} \frac{\partial }{\partial z} \left(\frac{z}{r^3}\right) = -\frac{p}{4 \pi \epsilon_0} \left(\frac{1}{r^3} - \frac{3z^2}{r^5} \right )$$

I'm not quite sure how he takes that derivative.

 

[jasonRF]

Note that $r = \sqrt{x^2 + y^2 + z^2}$. So insert $\sqrt{x^2 + y^2 + z^2}$ everywhere you see a $r$, take the derivative, and then
rewrite powers of $x^2 + y^2 + z^2$ in terms of $r$. Does that help?

jason

 

[jtbell]

Or apply the chain rule in the initial derivative to get an expression that contains $\partial r / \partial z$, then evaluate that derivative using $r = \sqrt{x^2 + y^2 + z^2}$, and finally rewrite the result completely in terms of $r$.