Development of the derivative of the error function

Notice that we've defined the squared error to be

$Error = \frac{1}{2} (y - \hat{y})^2.$

Also, we've defined the prediction to be

$\hat{y} = w_1 x + w_2.$

So to calculate the derivative of the Error with respect to
$w_1$
, we simply use the chain rule:

$\frac{\partial}{\partial w_1} Error = \frac{\partial Error}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial w_i}.$

The first factor of the right hand side is the derivative of the Error with respect to the prediction
$\hat{y}$, which is
$-(y-\hat{y}).$

The second factor is the derivative of the prediction with respect to
$w_1$, which is simply
$x$.

Therefore, the derivative is

Exercise

Calculate the derivative of the Error with respect to
$w_2$
and verify that it is precisely
$-(y-\hat{y}).$