Partial derivatives and total differentials

Question

Question 1: Partial Derivatives and Total Differentials (Show your steps!)

For f (x; y) = ln(x² + y), solve for its gradient vector and Hessian matrix.

Question 2: Chain Rule (Show your steps!)

For f (x, y, z) = ln(u + z), u = x²y + zy, solve for its total di§erential. You MUST use chain rule to solve for the partial di§erentials. (Note that u is not an argument of f (.
) while z is.)

Question 3: The Implicit Function Theorem (Show your steps!)

Let z = f (x, y) defined implicitly by the equation e^zy = xyz. Compute ∂z/∂x and ∂z/∂y :

Question 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 Eulerís Theorem is not satisfied.