Derivative of a vector with respect to a matrix
The quantity in question is a $3^{rd}$ order tensor.
One approach is to use index notation $$\eqalign{ f_i &= W_{ij} x_j \cr\cr \frac{\partial f_i}{\partial W_{mn}} &= \frac{\partial W_{ij}}{\partial W_{mn}} \,x_j \cr &= \delta_{im}\delta_{jn} \,x_j \cr &= \delta_{im}\,x_n \cr }$$ Another approach is vectorization $$\eqalign{ f &= W\,x \cr &= I\,W\,x \cr &= (x^T\otimes I)\,{\rm vec}(W) \cr &= (x^T\otimes I)\,w \cr\cr \frac{\partial f}{\partial w} &= (x^T\otimes I) \cr }$$