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

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In summary, the conversation is about finding a proper proof for the relation that u(x) multiplied by the partial derivative of u(x) with respect to x is equal to half of the partial derivative of u(x) squared with respect to x. The person asking for help has tried simple calculations but cannot find a mathematical proof. The person responding suggests using the product and chain rules for derivatives.
  • #1
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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  • #2
Did you ever encounter the product rule for derivatives? Or the chain rule? Both work.
 
  • #3
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]
 

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

What is the equation "Looking for a proof that u(x) du(x)/dx = 0.5 d(u(x)^2)/dx" trying to show?

The equation is trying to show that the derivative of a function u(x) multiplied by the derivative of u(x) is equal to half the derivative of u(x) squared.

Why is it important to find a proof for this equation?

Finding a proof for this equation is important because it is a fundamental rule in calculus and is used in many mathematical and scientific applications. It also helps to understand the relationship between derivatives and functions.

What are the steps to proving this equation?

The steps to proving this equation involve using basic calculus rules, such as the product rule and chain rule, to manipulate the equation and arrive at the desired result. It may also involve using algebraic manipulations and substitution of variables.

Are there any special cases or exceptions for this equation?

This equation holds true for all differentiable functions u(x). However, it may not hold true for non-differentiable functions or functions with discontinuities.

What are the practical applications of this equation?

This equation is commonly used in physics, engineering, and economics to find rates of change and optimize functions. It is also used in finding the area under a curve and determining the maximum or minimum values of a function.

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