U = Sin(x), then ∂U/∂x = What? The degree of the homogeneous function z = √(x + y) is? Evaluate partial derivatives and other expressions as given in the questions.

Steps to Solve

1. Find the Partial Derivative of U

To find the partial derivative of ( U = \sin(x) ) with respect to ( x ), we differentiate:

$$ \frac{\partial U}{\partial x} = \cos(x) $$

2. Determine the Degree of the Homogeneous Function

For the homogeneous function ( z = \sqrt{x + y} ), we need to find its degree. A function ( f(x, y) ) is homogeneous of degree ( n ) if it satisfies ( f(tx, ty) = t^n f(x, y) ). Here, we can see that:

$$ z(tx, ty) = \sqrt{tx + ty} = t\sqrt{x + y} $$

Thus, the degree is 1.

3. Calculate the Second Partial Derivative

For the function ( Z = f(x + ay) + \phi(x - ay) ), the second partial derivative ( \frac{\partial^2 Z}{\partial y^2} ) can be calculated as follows:

Differentiating once gives:

$$ \frac{\partial Z}{\partial y} = f'(x + ay) \cdot a + \phi'(x - ay)(-a) $$

Now differentiating again:

$$ \frac{\partial^2 Z}{\partial y^2} = f''(x + ay) \cdot a^2 + \phi''(x - ay)(-a^2) $$

4. Summary of Derivatives

The derivatives ( \frac{\partial^2 U}{\partial x^2} ) and ( \frac{\partial^2 Z}{\partial y^2} ) need to be calculated similarly for the other derivatives indicated in the document.