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Differentiation of integral

  1. Apr 6, 2010 #1
    Dear all, a question which has puzzled me for some days:
    (Assume that all are differentiable enough times):

    Calculate:
    [tex]
    \frac{\mathrm{d} }{\mathrm{d} x}\int_{g(x)}^{h(x)} f(x,t) dt
    [/tex]
     
  2. jcsd
  3. Apr 6, 2010 #2

    HallsofIvy

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    Leibniz's rule, which generalizes the fundamental theorem of Calculus:
    [tex]\frac{d}{dx} \int_{g(x)}^{h(x)} f(x,t)dt= \frac{dh}{dx}f(x, h(x))- \frac{dg}{dx}f(x,g(x))+ \int_{g(x)}^{h(x)} \frac{\partial f(x,t)}{\partial x} dt[/tex]
     
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