Find the equation of the tangent plane at the point $(1,1,1)$  to the conicoid $3x^2-y^2=2z$.

$3x^2-y^2=2z \implies 3x^2-y^2-2z=0 \equiv f(x,y,z)$
$\frac{\mathrm{d} f}{\mathrm{d} x} = 6x = 6 at x = 1$
$\frac{\mathrm{d} f}{\mathrm{d} y} = -2y = -2 at y = 1$
$\frac{\mathrm{d} f}{\mathrm{d} z} = -2$
This gives $6x - 2y - 2z = 6-2-2 = 2$. Thus, the equation of the tangent plane at given point is $6x - 2y - 2z = 2$