Find the equation of the plane tangent to the surface at

Question

X²y³z = 30 √(xyz + 2xy + 3z³ - 3) - 18       (2,3,4)

For each surface,

a. Find the equation of the plane tangent to the surface at the given point. Write the equation in the form z - z₀ = f_x(x₀,y₀)(x - x₀) + f_y(x₀,y₀)(y - y₀). Use implicit Differentiation to find f_x(x₀,y₀) and f_y(x₀,y₀). Do not use the shortcut formulas for f_x(x₀,y₀) and f_y(x₀,y₀) that are shown in part b.

b. Find the equation of the plane tangent to the surface at the given point. Write the equation in the form z - z₀ = f_x(x₀,y₀)(x - x₀) + f_y(x₀, y₀)(y-y₀). Use the shortcut formulas to find f_x(x₀,y₀) and f_y(x₀,y₀). That is, use f_x(x₀,y₀) = -F_x/F_z and f_y(x₀, y₀) = -F_y / F_z.

c. Find the equation of the plane tangent to the surface at the given point using gradients. Write the equation in the form ax + by + cx = d.

d. Compare parts (a) - (c) to make sure the equations are the same in all cases.