Via the chain rule, we can first take the partial of the relation with respect to [tex]\frac{\partial }{\partial x}\left ( x^2 + 4y^2 +9z^2 = 7)[/tex]
From the chain rule, we get:
[tex]\frac{\partial }{\partial x}\left ( x^2 + 4y^2 +9z^2 = 7 \right ) \to 2x +8y\frac{\partial y}{\partial x} + 18z\frac{\partial z}{\partial x} = 0[/tex]
Then solve for [tex]\frac{\partial z}{\partial x}[/tex]
[tex]\frac{\partial z}{\partial x}= -2x - 8y\frac{\partial y}{\partial x}[/tex]
The same can go for [tex]\frac{\partial z}{\partial y}[/tex] just take the partial with respect to y
