So this is a question that should be very easy but I can't do this. My knowledge of basic differentiation of partial derivatives aren't working. Can anyone help me how to do a and b. What I tried was the obvious take $r$ as the function of $x$ and $\theta$ and then differentiate, which doesn't work. Then I am thinking of doing implicit differentiation, which I'm not sure how to do based on the formulas I see. Thanks for your time.
We have $x= r \cos \theta$, $y= r \sin \theta$. We know that $r = \sqrt{x^2+y^2}$. Therefore $$\frac{dr}{dx} = \frac{1}{2\sqrt{x^2+y^2}}2x = \frac{x}{r} = \frac{r\cos\theta}{r} = \cos\theta.$$
You can solve the other tasks quite similarly.