Derivative of Integrals: Calculating with the Fundamental Theorem of Calculus

[EngWiPy]

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

 

[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.