Gradient Descent on Squared Error

Define error as:

$$E = \frac{1}{2} (T-O)^2 = \frac{1}{2} (T-g(\sum_j W_j I_j))^2$$

Then:

$$\frac{\partial E}{\partial W_j}=-I_j(T-O)g'(\sum_j W_j I_j) = -I_j \times Err \times g'(in)$$

And weights are updated a small amount in the opposite direction of the gradient:

$$W_j \leftarrow W_j + \alpha \times I_j Err \times g'(in)$$