Question

What is the partial derivative?

Answer 1

of U(x, y) = x^(1/2) * y^(1/2)

Answer 2

partial derivative of U with respect to x;and then the partial derivative with respect to y

Answer 3

Would it be: 1/2 *1/2y^(-1/2) FOR the partial derivative of x

Answer 4

When you take a partial derivative with respect to say, \(x\), then all the other variables are treated as constant. You're basically checking how the function behaves by changing only the value of a single variable in a given point.
So:

$$
U(x,y) = x^\frac{1}{2} \cdot y^\frac{1}{2} = \sqrt{x} \cdot \sqrt{y}
$$Now if \(y\) is constant then so is \(\sqrt{y}\) so:
$$
\frac{\partial U}{\partial x} = \sqrt{y} \cdot \frac{\partial (\sqrt{x})}{\partial x} = \sqrt{y} \cdot \frac{1}{2 \sqrt{x}}
$$
For reference you can see on the bottom:
http://www.wolframalpha.com/input/?i=U%28x%2Cy%29+%3D+x%5E%281%2F2%29+*+y%5E%281%2F2%29