# Cauchy Riemann equations - polar form question

Doing some exercises from a book and it says the following -

I dont get this. The bottom line differentiates both sides with respect to r and $ru_{r}$ becomes $ru_{rr} + ru_r$. Where is the $ru_{r}$ coming from, surely it should just be $ru_{rr}$?

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It's the product rule. – David Mitra Feb 12 '12 at 1:26

Recall $(fg)' =f'g+g'f$.
So \eqalign{{\partial\over \partial r} (r u_r)&= r {\partial\over \partial r} ( u_r)+u_r{\partial\over \partial r} (r ) \cr&=ru_{rr}+ u_r\cdot 1\cr &=ru_{rr}+ u_r .}
(By the way, you have a type in your second to last sentence. You should have "...and $ru_r$ becomes $ru_{rr}+ u_r$".)