In principle, the surface $z\left(x\,y\right)$ defined implicitly by the equation $h\left(x\,y\,z\right)\=0$ can be obtained explicitly by solving $h\=0$ for $z\=f\left(x\,y\right)$, in which case $h\left(x\,y\,f\left(x\,y\right)\right)\equiv 0$, that is, the equation becomes an identity in $x$ and $y$.

Implicit differentiation of this identity leads to

$h_x\+h_z f_x\=0\Rightarrow f_x\= -\frac{h_x}{h_z}$    and    $h_y\+h_z f_y\=0\Rightarrow f_y\= -\frac{h_y}{h_z}$ 

Use of $\mathrm{\λ}\=\sqrt{1\+f_x^2\+f_y^2}$, valid for the explicitly given surface $z\=f\left(x\,y\right)$, leads to

$\mathrm{\λ}\=\sqrt{1\+{\left(-\frac{h_x}{h_z}\right)}^2\+{\left(-\frac{h_y}{h_z}\right)}^2}$ = $\sqrt{\frac{h_x^{}\+h_y^2\+h_z^2}{h_z^2}}$ =  $\frac{\sqrt{h_x^{}\+h_y^2\+h_z^2}}{\left|h_z\right|}$ = $\frac{∥\nabla h∥}{\|h_z\|}$