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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question on derivative of multi variable.

**Physics Forums | Science Articles, Homework Help, Discussion**