# I Looking for a proof that u(x) du(x)/dx = 0.5 d(u(x)^2)/dx

1. Apr 25, 2017

### nonLinEul

Can anyone help with a proper proof for the following relation, please?

Code (Text):
$$u(x) \frac{\partial u(x)}{\partial x} = \frac{1}{2} \frac{\partial u(x)^2}{\partial x}$$
From simple calculations I agree that it's true, but it's been annoying me for a while that I can't find a proper mathematical proof for it.

See, for example (between equations 4 and 5): http://fluid.itcmp.pwr.wroc.pl/~znmp/dydaktyka/fundam_FM/Lecture9_10.pdf

Thanks!

2. Apr 25, 2017

### Orodruin

Staff Emeritus
Did you ever encounter the product rule for derivatives? Or the chain rule? Both work.

3. May 25, 2017

### phasacs

let u*du/dx=y <=> 2(u*du/dx)=2y <=> u*du/dx+u*du/dx=2y (use inverse product rule a'b+ab'= (ab)' )
<=> d/dx[u2]=2y <=> y=1/2 d/dx[u2]