If $w=f(x,y)$, where $x=r \cosθ$ and $y=r \sinθ$, then how to find $\dfrac{\partial}{\partial r}\left(\dfrac{\partial w}{\partial x}\right)$ and $\dfrac{\partial}{\partial r}\left(\dfrac{\partial w}{\partial y}\right)$.
I tried a lot but I didn't understand how to take $\dfrac{\partial}{\partial r}\left(\dfrac{\partial w}{\partial x}\right)$ and $\dfrac{\partial}{\partial r}\left(\dfrac{\partial w}{\partial y}\right)$.
If it's chain rule, then probably you mean: $\frac{\partial w}{\partial \theta}$ = $\frac{\partial w}{\partial x} \cdot\frac{\partial x}{\partial \theta} + \frac{\partial w}{\partial y} \cdot\frac{\partial y}{\partial \theta}$
Just compute the needed derivatives of $w, x, y$ functions and use them according to the formula above.