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

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The discussion centers on proving the mathematical relation u(x) ∂u(x)/∂x = 0.5 ∂(u(x)^2)/∂x. Participants confirm the validity of this equation through the application of the product rule and chain rule of derivatives. A detailed proof is provided, demonstrating that the left-hand side can be transformed into the right-hand side using established derivative rules. The reference to a specific lecture document supports the discussion's context and mathematical rigor.

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

Code: 
[tex] u(x) \frac{\partial u(x)}{\partial x} = \frac{1}{2} \frac{\partial u(x)^2}{\partial x}  [/tex]

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!
 
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Did you ever encounter the product rule for derivatives? Or the chain rule? Both work.
 
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]
 

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