#linear-layer

Let $W \in \mathbb{R}^{n \times m}$ have rows $\mathbf{w}_1^\top, \ldots, \mathbf{w}_n^\top$, let $\mathbf{v} \in \mathbb{R}^m$, and let $f: \mathbb{R} \to \mathbb{R}$ be differentiable. What is the gradient of $J(W) = \sum_{k=1}^n f(\mathbf{w}_k^\top \mathbf{v})$ with respect to $W$ (an $n \times m$ matrix)? Here $f'$ acts element-wise on the vector $W\mathbf{v}$.