My question is the related to the example below:
Why don't we use chain rule when differentiating this:
Example: Suppose $x,y$ are functions of $u,v$ and $z = x^2+y$, where
\begin{cases} x=e^u \cos(v) \\ y=e^u \sin(v) \end{cases}
Required to find: $$ \frac{\partial z}{\partial x} $$
If I'm differentiating $z=x^2+y$ with respect to $ x $, my first impulse would be to say
$$ \frac{\partial z}{\partial x} = 2x$$
But when I think about it, I don't understand why we don't use chain rule (when differentiating y) again because $ y $ and $x $ are related so isn't $y$ also a function of $ x$. $$ y = x \tan(v) $$
So why isn't the answer as below?: $$\frac{\partial z}{\partial x} = 2x+ \frac{dy}{dx} = 2x+\tan(v) $$
Can someone please clarify for me when chain rule is required and when it is not and please explain what is wrong with the logic/reasoning above. Thanks!