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I have this PDE \begin{align} u(x,t)\frac{\partial u(x,t)}{\partial t} &=u(x,t)\frac{\partial^2 u(x,t)}{\partial x^2} \quad \iff \\ \frac{1}{2}\frac{\partial}{\partial t}(u(x,t))^2&=u(x,t)\frac{\partial^2 u(x,t)}{\partial x^2} \end{align} How is the LHS rewritten? Why is $u(x,t)\frac{\partial u(x,t)}{\partial t} =\frac{1}{2}\frac{\partial}{\partial t}(u(x,t))^2$?

I guess I can't factor the $\partial/\partial t$, so $\frac{\partial}{\partial t}\big (u(x,t)u(x,t)\big )$?

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Hint: the Chain Rule.

$$ (f^2)'=2\,f\,f'. $$