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$u$ 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!

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You are correct, $$ \frac{\partial}{\partial y}(u u)= \frac{\partial}{\partial y}(u^2).$$ So letting $x=u^2$ we have $$ \frac{\partial}{\partial y}(u^2) = \frac{\partial x}{\partial y}=\frac{\partial x}{\partial u}\frac{\partial u}{\partial y}= 2u \frac{\partial u}{\partial y}.$$

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    Thanks! With $x=u^2$, do you mean $u$ is a function of two variables, $x=u(x,y)^2$? – 2017-02-19
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    $u=u(x,y)$ is a function of $x$ and $y,$ and $x=x(u)$ is a function of $u.$ So $x$ depends of $x$ and $y$ as well. So yes, $x=u^2=u(x,y)^2.$ – 2017-02-19