1. Dividing by powers

A simple question that has me tearing my hair out - how exactly does the dU/dX / dUdy given simplify to y/2x?

I have a feeling he makes a mistake in his differentiation but I still dont get how the simplification works!

2. Simple error

The first equation on the 2nd line, left should read:

$
\frac {\partial U} {\partial x} = \frac {1}{3} x ^ { -2/3} y ^ {2/3}
$

and the result follows.