Derivation of a variable limit integral

Question

Given $\mathbf{A}$ is $M\times M$ matrix, $\mathbf{a}$ and $\mathbf{b}$ are two $M\times 1$ column vector, and vector Gaussian distribution $\mathcal{N}(\mathbf{x}|\boldsymbol{\mu}, \mathbf{\Sigma})=\det(\pi \mathbf{\Sigma})^{-\frac{1}{2}}\exp \left[-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{\text{T}}\mathbf{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right]$, calculate the following partial derivation of variable limit integral \begin{align} \frac{\partial }{\partial \mathbf{x}}\int_{\mathbf{A}(\mathbf{a}-\mathbf{x})}^{\mathbf{A}(\mathbf{b}-\mathbf{x})}\mathcal{N}(\mathbf{t}|\mathbf{0},\mathbf{I})\text{d}\mathbf{t} \end{align} It seems that the result is $\mathbf{A}\left(\mathcal{N}(\mathbf{A}(\mathbf{b}-\mathbf{x})|\mathbf{0},\mathbf{I})-\mathcal{N}(\mathbf{A}(\mathbf{a}-\mathbf{x})|\mathbf{0},\mathbf{I})\right)$ which is $M\times M$ matrix. But, I believe the result should be $M\times 1$ column vector. So what's wrong?