Calculating the Gradient Descent

We can calculate the actual value of the gradient descent by differentiating $E$ : $\begin{matrix} \frac{\partial E}{\partial w_{i}} & = & \frac{\partial}{\partial w_{i}} \frac{1}{2} \sum_{d} (t_{d} - o_{d})^{2} \\ = & \frac{1}{2} \sum_{d} \frac{\partial}{\partial w_{i}} (t_{d} - o_{d})^{2} \\ = & \frac{1}{2} \sum_{d} 2 (t_{d} - o_{d}) \frac{\partial}{\partial w_{i}} (t_{d} - o_{d}) \\ = & \sum_{d} (t_{d} - o_{d}) \frac{\partial}{\partial w_{i}} (t_{d} - \overset{⇀}{w} \cdot \overset{⇀}{x_{d}}) \\ \frac{\partial E}{\partial w_{i}} & = & \sum_{d} (t_{d} - o_{d}) (- x_{i, d}) \end{matrix}$
So the weight update (training) rule is $Δ \overset{⇀}{w} = - η \sum_{d \in D} (t_{d} - o_{d}) x_{id}$ where $x_{id}$ is the single input component $x_{i}$ for example $d$ .