Let f(x, y, z) = z - arctan(x y). Compute the gradient of f at the point (0, 3, 0):
∇ f(x, y, z) = (-y / (1 + x²y²), -x / (1 + x²y²), 1)
∇ f (0, 3, 0) = (-3, 0, 1)
This vector is orthogonal to the surface z = f(x, y). Then the equation of the tangent plane is
∇ f (0, 3, 0) • (x, y - 3, z) = 0
(-3, 0, 1) • (x, y - 3, z) = 0
-3x + z = 0
z = 3x
