# Derivative of integrals

1. Nov 29, 2008

### S_David

What is the result of this derivative: $$\frac{d}{da}$$$$\int^{\infty}_{a} f_{1}(ax)f_{2}(x)dx$$

2. Nov 29, 2008

### adriank

You can write, say, $$g(a, b) = \int_a^\infty f_1(bx)f_2(x) \,dx$$; then using the chain rule you get $$\frac{d}{da} g(a, a) = g_1(a, a) + g_2(a, a)$$, 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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