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Homework Help: Partial derivative - why is it count like this?

  1. Dec 19, 2009 #1
    1. The problem statement, all variables and given/known data
    I have John R Taylor "Classical mechanics" part 1, and I have an integral:
    [tex]\int\limits^{x_2}_{x_1}f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)\mbox{d}x[/tex]
    and here is count derivative of underintegral function in [tex]\alpha[/tex]
    [tex]\frac{\partial f\left(y+\alpha\eta,y^{\prime}+\alpha\eta^{\prime},x\right)}{\partial\alpha}=\eta\frac{\partial f}{\partial y}+\eta^{\prime}\frac{\partial f}{\partial y^{\prime}}[/tex]
    why there is suddenly [tex]\frac{\partial f}{\partial y^{\prime}}[/tex] and [tex]\frac{\partial f}{\partial y}[/tex], while this derivative is by [tex]\alpha[/tex] - why not only [tex]\eta,\eta^{\prime}[/tex]?
    2. Relevant equations
    I was thinking about function composition derivative, but it didnt helped me.
    3. The attempt at a solution
    Nothing, I couldnt do anything with this, I dont know why this is count like this, please help;] thanks!
     
  2. jcsd
  3. Dec 19, 2009 #2
    [tex]f=f(y,y',x)[/tex]

    Now if you replace [tex]y=\bar{y}+\alpha\eta[/tex] then y becomes a function of alpha, which means that you have to use the chain rule.

    Try choosing an explicit non-linear expression for f if that makes it more clear. For example f(y,y',x) = y2 + y'2
     
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