Multiplying out brackets with partial derivitives.

[AStaunton]

in my notes I have the following:

in my notes I have:

$$\cos\theta\frac{\partial}{\partial r}(\cos\theta\frac{\partial u}{\partial r}-\frac{\sin\theta}{r}\frac{\partial u}{\partial\theta})-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}(\cos\theta\frac{\partial u}{\partial r}-\frac{\sin\theta}{r}\frac{\partial u}{\partial\theta})$$

$$=\cos^{2}\theta\frac{\partial^{2}u}{\partial r^{2}}-\frac{\sin\theta\cos\theta}{r}\frac{\partial^{2}u}{\partial r\partial\theta}+\frac{\sin\theta\cos\theta}{r^{2}}\frac{\partial u}{\partial\theta}-\frac{\sin\theta\cos\theta}{r}\frac{\partial^{2}u}{\partial\theta\partial r}+\frac{\sin^{2}\theta}{r}\frac{\partial u}{\partial r}+\frac{\sin\theta\cos\theta}{r^{2}}\frac{\partial u}{\partial\theta}+\frac{\sin^{2}\theta}{r^{2}}\frac{\partial^{2}u}{\partial^{2}\theta}$$


I cannot figure out where the two partial(u)/partial(theta) expressions came from and also where did the partial(u)/partial(r) expression come from?
I clearly don't understand the rules of this properly, my thinking was that we get rid of the brackets by multiplying everything out, but that does not account for the expressions I just mentioned..

 

[Tangent87]

You don't multiply everything out, you differentiate everything out. So for example the \partial theta will differentiate the cos@ and give you a partial(u)/partial(r) term.