Looks like the given curve is
5xy + z⁴x - 3yz = 3
Differentiate both sides with respect to x :
5y + 4z³x ∂z/∂x + z⁴ - 3y ∂z/∂x = 0
Solve for ∂z/∂x :
(4z³x - 3y) ∂z/∂x = - (5y + z⁴)
∂z/∂x = - (5y + z⁴) / (4z³x - 3y)
At the point (1, 1, 1), this derivative is
∂z/∂x (1, 1, 1) = - (5 + 1) / (4 - 3) = -6
