Finding the partial derivatives of $f(x,y,2x^2+ y^2) = 2x +5y$

Question

So, given that $f(x,y,2x^2+ y^2) = 2x + 5y$,

I chose $z(x,y) = 2x^2 + y^2$, so that $f(x, y, z(x,y)) = 2x + 5y$.

I just can't figure out the partial derivatives for $x$, $y$, and $z$ of $f$.

What I get -

for $f_x$ -

$f_x + f_z * z_x = 2x$ (I just tried to differentiate the whole equation).

but what is $f_z$?

help :(

Answer 1

Your implicit equation only defines $f$ on the surface $z = 2x^2 + y^2$. $f_z(x_0,y_0,z_0)$ requires that $f$ be defined at least in a subset of the form $\{(x_0,y_0,z):z\in(z_0-\delta,z_0+\delta)\}$.