I am wondering about differentiation under the integral sign when the integral in question is actually over positive-definite matrices using the Haar invariante measure. Suppose we have \begin{eqnarray} I(Y) = \int_{X>0} f(X,Y) (dX) \end{eqnarray} where $f$ is a well-behaved continuous, differentiable function of two (or more) arguments. Suppose I wish to find $\frac{dI}{dY}$. We can also assume that $f$ is some analytic function like the trace or the exponential trace, those common in multivariate analysis.
Can someone point me to a standard reference or some formula in the style of http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign or perhaps show me how to arrive at this ?
Thanks so much!