Differentiation with respect to a parameter

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Use a parameter when differentiating it simplifies the integrand. Justify interchanging derivative and integral, solve and recover parameter dependence from a known condition. Parameter-dependent bounds add Leibniz boundary terms.

Step-by-step example

1. Take a>0 and evaluate the base integral.

   $$F(a)=\int_0^\infty e^{-ax}\,dx=1/a$$

2. Local domination on a>0 permits differentiation.

   $$F'(a)=-\int_0^\infty xe^{-ax}\,dx=-1/a^2$$

3. Isolate the desired integral.

   $$\int_0^\infty xe^{-ax}\,dx=1/a^2$$