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Homework Help: Please just check this [Analysis problem]

  1. Nov 10, 2004 #1

    AKG

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    Find the (first-order) partial derivatives of the following function (where g : [itex]\mathbb{R} \to \mathbb{R}[/itex] is continuous):

    [tex]f(x, y) = \int _a ^{\int _b ^y g} g.[/tex]

    --------------

    I got:

    [tex]D_1f(x, y) = 0[/tex]

    [tex]D_2f(x, y) = \frac{\partial f}{\partial y}[/tex]

    [tex]= \frac{\partial }{\partial y} \int _a ^{\int _b ^y g}g[/tex]

    Let G be the antiderivative of g:

    [tex]= \frac{\partial }{\partial y}\left ( G \left (\int _b ^y g\right ) - G(a) \right )[/tex]

    [tex]= \frac{\partial }{\partial y} G \left (\int _b ^y g\right )[/tex]

    [tex]= \left ( \frac{\partial G \left (\int _b ^y g\right )}{\partial \left (\int _b ^y g\right ) }\right ) \left (\frac{\partial }{\partial y}\int _b ^y g\right )[/tex]

    [tex]= g\left (\int _b ^y g\right )\frac{\partial }{\partial y}\left (G(y) - G(b) \right )[/tex]

    [tex] = g\left (\int _b ^y g\right )g(y)[/tex]
     
    Last edited: Nov 10, 2004
  2. jcsd
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