The problem says: If the equation $x^2 +y^2 +z^2 = G(ax+by +cz)$ defines $z=f(x,y)$, $f$ and $G$ being differentiable, $a$, $b$ and $c$ constants, find $\partial z/\partial x$.
Is this correct?:
$$ \frac{\partial z}{\partial x} = -\frac{Gx}{Gz} = - \frac{2ax}{2cz}. $$
I'm kind of confused with the constant as they are not in $x^2 +y^2 +z^2$.
The problem suggests $x^2 + y^2 + z^2 =\text{const}$ and $G(ax+by+cz)=\text{const}$ define two surfaces respectively. Their intersections, defined by $x^2 + y^2 + z^2 = G(ax+by+cz)$, is a line $z=f(x,y)$. Here, we consider $x,y$ to vary independently, meaning $\partial y/\partial x = 0$.
Back to the problem, we define the line equation as $$ x^2 + y^2 + z^2 - G(ax+by+cz) = 0. $$ And differentiating with respect to $x$ gives $$ \begin{align} & 2x + 0 + 2 z \frac{\partial z}{\partial x} - \frac{\partial G(ax+by+cz)}{\partial x} = 0 \\ \Rightarrow \quad & 2x + 2 z \frac{\partial z}{\partial x} - G'(\gamma) \left.\left(a+c\frac{\partial z}{\partial x}\right)\right|_{\gamma=ax+by+cz} =0 \\ \Rightarrow \quad & \frac{\partial z}{\partial x} = \left.\frac{aG'(\gamma)-2x}{-cG'(\gamma)+2z} \right|_{\gamma=ax+by+cz}. \end{align} $$