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?
Why is $uu_x = ( \frac12 u^2 )_x$?
0
$\begingroup$
pde
partial-derivative
2 Answers
5
This is chain rule
$$\frac{d}{dx}u(x)^2=2u(x)u'(x)$$
2
Its just an application of the chainrule. Taking $f(x)=\frac{1}{2}x^2$ and looking at $f(u(x))$ you get: $$ (f(u))_x=f_x(u) \cdot u_x=\frac{1}{2}\, 2\, u\cdot u_x=uu_x $$