$\mathbf X = [\mathbf x_1|\mathbf x_2|...|\mathbf x_N]$ is an $M$-by-$N$ matrix, where $\mathbf x_n$ is a column vector $\in \mathbf R^M$, and $n=1,2,...,N$.
Given an $M$-by-1 column vector $\mathbf b$, $\mathbf X^{\top} \mathbf b$ can be written as follows, i.e., $\mathbf X^{\top} \mathbf b= \mathbf I_{:,1} \mathbf x_1^{\top} \mathbf b + ... + \mathbf I_{:,N} \mathbf x_N^{\top} \mathbf b$, where $\mathbf I$ is an $M$-by-$M$ identity matrix, and ":" means selecting all the rows.
I am wondering if $\frac{\partial \mathbf X^{\top} \mathbf b}{\partial \mathbf x_n}=\mathbf I_{:,n}\mathbf b^{\top}$ because $\frac{\partial \mathbf X^{\top}}{\partial \mathbf x_n}$ is a tensor.