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How to differentiate an integral?

  1. Mar 5, 2009 #1
    I just need a short reminder from Calculus. Suppose you have a linear functional [tex]\alpha[/tex] from C1[-1,1] to [tex]\Re[/tex], given by

    [tex]\alpha(f) = \int_{-1}^{1}f(t)g(t)dt[/tex]

    for some fixed continuous function g. What is [tex]\frac{d \alpha}{d f}[/tex]?
  2. jcsd
  3. Mar 6, 2009 #2


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    [tex]\frac{d \alpha}{d f} = \int_{-1}^{1}\frac{d}{df}f(t)g(t)dt [/tex]
  4. Mar 6, 2009 #3
    Is there any way to simplify (or expand) this?

    I am tempted to think that

    [tex]\int_{-1}^{1}\frac{d}{df}f(t)g(t)dt = \int_{-1}^{1}g(t)dt [/tex]

    but that cannot be right because it's just a constant and integrating it would give me [tex]f[/tex] times a constant instead of an integral involving [tex]f[/tex] .
    Last edited: Mar 6, 2009
  5. Mar 6, 2009 #4


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  6. Mar 6, 2009 #5
    Thanks a lot!
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