Gradients of loss functions

Similarly, the gradients of \( \mathcal{L} \) with respect to the weights and biases follow,

$$ \begin{align*} \nabla_{\mathbf{c}} \mathcal{L} &=\sum_{t}\left(\frac{\partial \mathbf{y}^{(t)}}{\partial \mathbf{c}}\right)^\mathsf{T} \nabla_{\mathbf{y}^{(t)}} \mathcal{L} \notag\\ \nabla_{\mathbf{b}} \mathcal{L} &=\sum_{t}\left(\frac{\partial \mathbf{h}^{(t)}}{\partial \mathbf{b}}\right)^\mathsf{T} \nabla_{\mathbf{h}^{(t)}} \mathcal{L} \notag\\ \nabla_{\mathbf{V}} \mathcal{L} &=\sum_{t}\sum_{i}\left(\frac{\partial \mathcal{L}}{\partial y_i^{(t)} }\right)\nabla_{\mathbf{V}^{(t)}}y_i^{(t)} \notag\\ \nabla_{\mathbf{W}} \mathcal{L} &=\sum_{t}\sum_{i}\left(\frac{\partial \mathcal{L}}{\partial h_i^{(t)}}\right)\nabla_{\mathbf{w}^{(t)}} h_i^{(t)} \notag\\ \nabla_{\mathbf{U}} \mathcal{L} &=\sum_{t}\sum_{i}\left(\frac{\partial \mathcal{L}}{\partial h_i^{(t)}}\right)\nabla_{\mathbf{U}^{(t)}}h_i^{(t)}. \tag{1} \end{align*} $$