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Derivative of integrals

  1. Nov 29, 2008 #1
    What is the result of this derivative: [tex]\frac{d}{da}[/tex][tex]\int^{\infty}_{a} f_{1}(ax)f_{2}(x)dx[/tex]
  2. jcsd
  3. Nov 29, 2008 #2
    You can write, say, [tex]g(a, b) = \int_a^\infty f_1(bx)f_2(x) \,dx[/tex]; then using the chain rule you get [tex]\frac{d}{da} g(a, a) = g_1(a, a) + g_2(a, a)[/tex], where g1 is the partial derivative of g with respect to the first argument, and similarly for g2. For calculating g1, use the fundamental theorem of calculus; for calculating g2, move the derivative under the integral sign.
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