I want to calculate the derivative of a density function with respect to its parameters. In my case the density is given by: $$ p( x | y )= \int p( x | y, z ) p( z | y ) ~dz $$ where $ z|y\sim \mathcal{N}( \mu(\theta,y), \sigma(\theta,y) ) $ and $\theta$ is a constant parameter.
I'm looking for the derivative of $p( x | y )$ with respect to $\theta$ $$ d/d\theta~p( x | y ) = d/d\theta \left( \int p( x | y, z ) p( z | y ) ~dz \right) $$ My approach was as follows (using chain rule): $$ d/d\theta~p( x | y ) = d/dz \left( \int p( x | y, z ) p( z | y ) ~dz \right) \cdot dz/d\theta $$
but left me with $dz/d\theta $ which I don't know how to interpret and therefore made me questioning the full approach. What am I doing wrong here?