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I think your notations are unclear, but I can give an answer based on what you probably meant. For example, $\frac{\partial{L}}{\partial{W^x}}$ should be replaced by $(\nabla_{W^x_{j:}}L)_{j=1, ...,n}$ (assuming everything stays in $\mathbb{R}^n$). Also your expression for $\frac{\partial{L}}{\partial{W^x}}$ is wrong, even accounting for the notation. Since \$...