How to find a directional derivative of an implicit function? it's not my homework, I just want to find out how to find a directional derivative of an implicit function. I know what is a directional derivative and how to find it when I have a function in normal form (I mean like like z=x^2+y....).
$$
xz + yz^2 = 3xy + 3
$$
the point is:
$$
P(1,−1)
$$
and the direction (vector):
$$
u = [1, 1]
$$
Could you give me a formula for this? I know how to compute the derivatives of an implicit function as well.

Implicit differentiation (officially, the Implicit Function Theorem if you rewrite your equation as $F(x,y,z)=0$) allows you to compute the gradient of $z=g(x,y)$, and then you use your usual dot product. 
