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Math Help - Derivative of a definite integral

  1. #1
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    Derivative of a definite integral

    What's equivalent to the following expression?

    \frac { \partial} {\partial p}\int _p^{p_0} f(\alpha,\beta)d\alpha
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by altave86 View Post
    What's equivalent to the following expression?

    \frac { \partial} {\partial p}\int _p^{p_0} f(\alpha,\beta)d\alpha
    Assume that \int f(\alpha,\beta)\,d\alpha = F(\alpha,\beta). Thus,

    \int_p^{p_0}f(\alpha,\beta)\,d\alpha = F(\alpha,\beta)\big|_p^{p_0}=F(p_0,\beta)-F(p,\beta)

    Note that all operations you are doing are either w.r.t. \alpha or p, so \beta and p_0 should be treated as constants, and therefore F(p_0,\beta) is a constant. Call it C.

    Now we want \frac{\partial}{\partial p}[C-F(p,\beta)]=\frac{\partial}{\partial p}[-F(p,\beta)] because the derivative of a constant is zero.

    Remember how we defined F though. We defined it so that F~'=f. So

    \frac{\partial}{\partial p}[-F(p,\beta)]=\boxed{-f(p,\beta)}
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