I am a bit stuck with the notation and, thus, the calculations. I need to find the partial derivative of z with respect to x, keeping r constant.
So my solution would be: $(\frac{\partial x}{\partial x})_{r} = (\frac{\partial z}{\partial r}) \cdot (\frac{\partial r}{\partial x}) + (\frac{\partial z}{\partial \theta}) \cdot (\frac{\partial \theta}{\partial x})$
But because r is constant, $(\frac{\partial r}{\partial x}) = 0$ and we are left with $(\frac{\partial z}{\partial \theta}) \cdot (\frac{\partial \theta}{\partial x})$
Substituting $x = rcos(\theta)$ and $y = rsin(\theta)$
$(\frac{\partial x}{\partial x})_{r} = - \frac{sin(\theta)}{r} \cdot (2r^2sin(\theta)cos(\theta) + 4r^2sin(\theta)cos(\theta)) = -6rsin^2(\theta)cos(\theta)$
Which is incorrect because the answer should be $-6x$.
Could you please tell me where is a mistake in my reasoning?