I am having troubles finding partial derivatives.
If $f(x,y)=2x^2+y^2$ then, $$f_x=4x$$ $$f_y=2y$$
That's simple enough. But when I see a $z$ in the equation, I get stumped. I know $z=f(x,y)$. I don't really see the process.
For example, if $z=2x^2+y^2$ then do we differentiate both sides with respect to $x$ like this?
Even worse, what am I supposed to do for something like this? Same thing?
I'm pretty sure that's wrong but I don't know why. Can someone please help me understand? Thanks.
Edit Some context for the last example (it's from a homework problem):
Find the equation of the tangent plane to the surface with equation $x+y^2+z^3=3$ at the point (2,1,0).
I know that the equation for the tangent plane is $$z=f(x_0,y_0)+[f_x(x_0,y_0)](x-x_0)+[f_y(x_0,y_0)](y-y_0)$$
Since the $f_x$ and $f_y$ found above contain z, do I plug in $z_0$?
So I'm trying to find $f_x$ and $f_y$.
I get it now