1. The problem statement, all variables and given/known data "Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g, differentiable. (a) What are the derivatives f'(1), g'(1)? (b) What is the tangent line to the curve of intersection (1,1,1) in the forms x = 1 + ___ s, y = 1 + ___ s, z = 1+s" 2. Relevant equations Formula for gradient. 3. The attempt at a solution For the last part, I think I see that t = 1 + s but I'm not sure. For the first part, I computed the gradients and tried (and failed) to equate them since the surfaces are intersecting. gradF(x,y,z) = (4x^3 + 3yz) i + (2y - 3xz) j + (6z^5 - 3xy) k gradG(x,y,z) = (y+z) i + (x+z) j + (y+x - 24z^7) k The correct answers are: f'(1) = 4, g'(1) = 7, x = 1 + 4s, y = 1 + 7s. Any help in solving this problem would be greatly appreciated! Thanks in advance!