(4A-1) This curve is the intersection of the function z = x2 + 2y2 with the plane x = -0.5.
① substituting in the fixed value of x, we get: z = 0.25 + 2y2
② Differentiating, we get: dz/dy = 4y. So at y = -0.5: dz/dy = -2.
This is the gradient of the tangent.
③ z = x2 + 2y2, so ∂z/∂y = 4y. This is the same as we got in ② because the partial derivative is defined as the derivative with respect to y when x is constant, which is - obviously - the case when we define x = -0.5