Weight Update Rule: This equation is updating the weight based on its previous value and the change in weight.     $v_i(t) = v_i(t-1) + \Delta v_i(t)$    -  $v_i(t)$ is the weight of the connection from input unit $i$ to the neural network at time  $t$.    -  $t$ is the current iteration or time step.    -  $\Delta v_i(t)$ is the change in the weight at time $t$.

Change in Weight ( $\Delta v_i(t)$): $v_i(t) = \eta \left(-\frac{\partial \mathcal{E}}{\partial v_i}\right)$    -  $\Delta v_i(t)$ is the change in weight at time $t$.    -  $\eta$ is the learning rate, controlling the size of the weight updates.    -  $-\frac{\partial \mathcal{E}}{\partial v_i}$ is the negative gradient of the error ($\mathcal{E}$) with respect to the weight $v_i$.    - This term indicates the direction and magnitude of the change needed to minimize the error.

Gradient Calculation: $\frac{\partial \mathcal{E}}{\partial v_i} = -2\left(t_p-o_p\right) \frac{\partial f}{\partial \text { net }_p} z_{i, p}$    -  $\frac{\partial \mathcal{E}}{\partial v_i}$ is the gradient of the error with respect to the weight  $v_i$.    -  $t_p$ is the target output, $o_p$ is the actual output, and  $\mathcal{E}$ is the error.    -  $\frac{\partial f}{\partial \text { net }_p}$ is the derivative of the activation function $f$ with respect to the net input to the output unit.    -  $z_{i, p}$ is the input from unit $i$ for pattern $p$.