derivative of polar coordinates and difference between partial to normal

Question

So there's the task and how I did it, but I have the feeling it's not correct. Can someone help with the difference between a and b and how to do all of them...

I'm also not sure what to do with these polar coords... :/

First task: $(a)$ Given $L = t^2 + x^2(t) + y(t)$, with $x(t) = \sin(2t)$ and $y(t) = \cos(t)$, calculate the following:

$\hspace{1cm} a) \frac{dL}{dt}$

$\hspace{1cm} b)\frac{\partial L}{\partial t}$

$\hspace{1cm} c) \frac{\partial L}{\partial x}$

$\hspace{1cm} d) \frac{\partial L}{\partial y}$

second: Call $(x,y)$ and $(r,θ)$ the Cartesian and Polar coordinates, related in the usual way. Given $L = x^2(r,θ) + y^2(r,θ)$, calculate

$\hspace{1cm} a) \frac{dL}{dr}$

$\hspace{1cm} b) \frac{dL}{d\theta}$

I interpert the $2nd$ one as $L= r^2\cos^2+r^2\sin^2 = r^2$ then the derivative are easy, but I am not sure if that's the correct $L$ - what does it mean $x^2(r,θ)$?...

Thanks!