$u$ is a function of two variables, $u(x,y)$.
How can I use the chain rule to write $ u\frac{\partial u}{\partial y} $ as $\frac{1}{2}\frac{\partial u^2}{\partial y}$?
Is it correct to write $$ \frac{\partial}{\partial y}\big(u u\big )= \frac{\partial}{\partial y}\big(u^2\big ) \quad \text{?} $$
Thanks!
You are correct, $$ \frac{\partial}{\partial y}(u u)= \frac{\partial}{\partial y}(u^2).$$ So letting $x=u^2$ we have $$ \frac{\partial}{\partial y}(u^2) = \frac{\partial x}{\partial y}=\frac{\partial x}{\partial u}\frac{\partial u}{\partial y}= 2u \frac{\partial u}{\partial y}.$$