(4A-2)   The full function is   z = x2 +  2y

In this instance, y is constrained to be equal to -0.5. Therefore the function simplifies to:

   z = x2 + 0.5.

The gradient of the tangent is given by the derivative dz/dx = 2x.

When x = -0.5, the gradient is therefore equal to -1.

When, as in this case, the value of y is held constant the partial derivative of the function z = x2 +  2y2, ∂z/∂x = 2x. This is, as it must be, the same as the derivative of the function formed by the intersection of the z = x2 + 0.5 surface with the plane y = -0.5  (ie y is constant).