Differentiation Under the Integral Sign
Another question about “partial integrals” is when we can interchange integral and differential operations. Just like we found for continuity of partial integrals, this involves interchanging two limit processes.
Specifically, let’s again assume that is a function on the rectangle , and that is of bounded variation on and that the integral
exists for every . Further, assume that the partial derivative exists and is continuous throughout . Then the derivative of exists and is given by
Similar results hold where and are vector values, and where derivatives in terms of are replaced outside the integral by partial derivatives in terms of its components.
So, if and we can calculate the difference quotient:
where is some number between and that exists by the differential mean value theorem. Now to find the derivative, we take the limit of the difference quotient
as we take to approach , the number gets squeezed towards as well. Since we assumed to be continuous, the limit in the integrand will equal , as asserted.