I presume that the letter after the derivative indicates that the derivative is to be taken holding that variable constant.
So here is the idea for the first few:
1) \(\displaystyle \left(\frac{\partial z}{\partial x}\right)_y\)
Since \(\displaystyle z= x^2+ 2y^2\). Holding y constant, the derivative with respect to x is 2x.
2) \(\displaystyle \left(\frac{\partial z}{\partial x}\right)_r\)
\(\displaystyle x^2+ y^2= r^2cos^2(\theta)+ r^2 sin^2(\theta)= r^2(cos^2(\theta)+ sin^2(\theta))= r^2\) so \(\displaystyle y^2= r^2- x^2\)
\(\displaystyle x^2+ 2y^2= x^2+ 2(r^2- x^2)= -x^2+ 2r^2\). Holding r constant, the derivative with respect to x is -2x.
3) \(\displaystyle \left(\frac{\partial z}{\partial x}\right)_\theta\)
\(\displaystyle \frac{y}{x}= \frac{r sin(\theta)}{r cos(\theta)}= tan(\theta)\) so \(\displaystyle y= x tan(\theta)\) and \(\displaystyle x^2+ 2y^2= x^2+ 2x^2 tan^2(\theta)= x^2(1+ 2 tan^2(\theta))\). Holding \(\displaystyle \theta\) constant the derivative with respect to x is \(\displaystyle 2x(1+ 2tan^2(\theta))\)
Now, YOU try the others!