Use implicit differentiation to find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}$

$7x^{2}z+2z^4-6yz=0$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=-\frac{7x^2+8z^3-6y}{14xz}$;$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=\frac{7x^2+8z^3-6y}{6z}$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=\frac{7x^2+8z^3-6y}{14xz}$;$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=-\frac{7x^2+8z^3-6y}{6z}$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=\frac{14xz}{7x^2+8z^3-6y}$;$1\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=-\frac{6z}{7x^2+8z^3-6y}$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=-\frac{14xz}{7x^2+8z^3-6y}$;$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=\frac{6z}{7x^2+8z^3-6y}$