#vjp

A layer $g$ maps $x \in \mathbb{R}^2$ to $y \in \mathbb{R}^3$. At the current point its Jacobian (rows = outputs, $J_{ji} = \partial y_j/\partial x_i$) and the upstream gradient are

$J = \begin{bmatrix} 1 & 2 \\ 0 & -1 \\ 3 & 1 \end{bmatrix},\qquad \nabla_y L = \begin{bmatrix} 1 & 2 & -1 \end{bmatrix}.$

The vector–Jacobian product gives $\partial L/\partial x_1 =$ and $\partial L/\partial x_2 =$ .