Question

Find the value of StartFraction partial derivative z Over partial derivative x EndFraction at the point​ (1,1,1) if the equation 5 xy plus z Superscript 4 Baseline x minus 3 yz equals 3 defines z as a function of the two independent variables x and y and the partial derivative exists.

Answer 1

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