Error Gradient for Sigmoid Unit

Since the networks have multiple output units we must redefine $E$ to sum the errors $E (\overset{⇀}{w}) \equiv \frac{1}{2} \sum d \in D \sum_{k \in outputs} (t_{kd} - o_{kd})^{2}$ where $t_{kd}$ and $o_{kd}$ are the target and output values associate with the kth output unit and example $d$ .
After some algebra (not shown) we can derive that the error slope for each weight is $\frac{\partial E}{\partial w_{i}} = - \sum_{d \in D} (t_{d} - o_{d}) o_{d} (1 - o_{d}) x_{i, d}$