1. Partial Derivatives and Total Di造erentials (Show your steps!) For f (x; y) = ln(x2 + y), solve for its gradient vector and Hessian matrix. 2. Chain Rule (Show your steps!) For f (x; y; z) = ln(u+z), u = x2y +zy, solve for its total di造erential. You MUST use chain rule to solve for the partial di造erentials. (Note that u is not an argument of f () while z is.) 3. The Implicit Function Theorem (Show your steps!) Let z = f (x; y) dened implicitly by the equation ezy = xyz. Compute @z @x and @z @y : 4. Homogenous Function (Show your steps!) Is f (x; y; z) = x2y2z2 a homogenous function or not? If so, solve for its degree. If not, prove it by showing the Eulers Theorem is not satised. 1