Why is $uu_x = ( \frac12 u^2 )_x$?

Question

I see that this is true, by doing some examples:

For instance, if $$u = x^2$$ we have,

$$uu_x = 2x^3$$ $$( \frac12 u^2 )_x = 2x^3$$

How can we manipulate one side to show it is equal to the other side?

Answer 1

This is chain rule

$$\frac{d}{dx}u(x)^2=2u(x)u'(x)$$

Answer 2

Its just an application of the chainrule. Taking $f(x)=\frac{1}{2}x^2$ and looking at $f(u(x))$ you get: $$ (f(u))_x=f_x(u) \cdot u_x=\frac{1}{2}\, 2\, u\cdot u_x=uu_x $$