I have a hard time taking the derivative [tex] \frac{\partial u}{\partial A} [/tex](adsbygoogle = window.adsbygoogle || []).push({});

Where [tex] u=A+B[/tex] and [tex]A = x+2y,B = x-2y [/tex]

This is because A and B are both function of x and y. I don't think I can treat A and B as totally indenpend to each other where

[tex] \frac{\partial B}{\partial A}=0, \frac{\partial A}{\partial B}=0 [/tex] .

This is what I did and please tell me what did I do wrong.

[tex]A = x+2y,B = x-2y \Rightarrow x=\frac{A +B}{2}, y=\frac{A - B}{4} [/tex]

[tex]\Rightarrow \frac{\partial A}{\partial x}= 1 , \frac{\partial A}{\partial y}= 2 , \frac{\partial B}{\partial x}= 1 , \frac{\partial B}{\partial y}= -2[/tex]

[tex]x=\frac{A +B}{2}\Rightarrow \frac{\partial x}{\partial A}=\frac{1}{2} [\frac{\partial A}{\partial A}+\frac{\partial B}{\partial x}\frac{\partial x}{\partial A}] = \frac{1}{2} [1 + \frac{\partial B}{\partial x}\frac{\partial x}{\partial A}] = \frac{1}{2} + \frac{1}{2}\frac{\partial x}{\partial A}\Rightarrow \frac{\partial x}{\partial A} = 1[/tex]

[tex]y=\frac{A - B}{4} \Rightarrow \frac{\partial y}{\partial A}=\frac{1}{4} [\frac{\partial A}{\partial A}- \frac{\partial B}{\partial y}\frac{\partial y}{\partial A}] = \frac{1}{4} [1 + 2\frac{\partial y}{\partial A}] = \frac{1}{4} + \frac{1}{2}\frac{\partial y}{\partial A}\Rightarrow \frac{\partial y}{\partial A} = \frac{1}{2}[/tex]

[tex]\frac{\partial y}{\partial B}=\frac{1}{4} [\frac{\partial A}{\partial y}\frac{\partial y}{\partial B} - \frac{\partial B}{\partial B}] = \frac{1}{4} [ 2\frac{\partial y}{\partial B} - 1] = \frac{1}{2}\frac{\partial y}{\partial B} -\frac{1}{4} \Rightarrow \frac{\partial y}{\partial B} = -\frac{1}{2}[/tex]

[tex]\frac{\partial u}{\partial A} = \frac{\partial A}{\partial A}+ \frac{\partial B}{\partial x}\frac{\partial x}{\partial A} + \frac{\partial B}{\partial y}\frac{\partial y}{\partial A} = 1 +\frac{\partial B}{\partial x} + \frac{1}{2}\frac{\partial B}{\partial y} = 0[/tex]

Obviously I did it wrong, can anyone tell me what I did wrong. This is not homework. Question is how to treat A and B when they both are function of x & y.

Thanks

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# Question on derivative of multi variable.

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