I am trying to take the derivative of the multivariate cumulative normal distribution wrt a certain parameter x. Both the mean (Hx, where H is a n x 1 vector) and the variance (n x n matrix T) of my normal distribution depend on x.
$$\frac{\int_{t=-\infty}^{t=\mathbf{H}x} \vert \mathbf{T}_i \vert^{-1/2}\frac{1}{(2\pi)^{n/2}} e^{-.5t'\mathbf{T}^{-1}_it}dt}{\partial x}$$
However, when I want to apply the Leibniz rule, I first take the derivative of the upper bound (a vector wrt a scalar) and I end up with a n x 1 vector. Next, I take the derivative of my normal distribution, and I end up with a n x n matrix. What is my mistake?