使用链式法则，

$$\frac{\partial C}{\partial W_1} = \frac{\partial C}{\partial z} \frac{\partial z}{\partial h} \frac{\partial h}{\partial a} \frac{\partial a}{\partial W_1} ,$$ 其中
$$a = W_1x+b_1 .$$

$$\frac{\partial C}{\partial z} = y-\hat{y},$$

$$\frac{\partial z}{\partial h} = W_2,$$

$$\frac{\partial h}{\partial a} = \sigma(a)(1 - \sigma(a)),$$

$$\frac{\partial a}{\partial W_1} = x,$$ 所以 $$\frac{\partial C}{\partial W_1} = (y-\hat{y})W_2\sigma(a)(1 - \sigma(a))x = (y-\hat{y})W_2h(1 - h)x.$$

有关详细信息，您可以查看模式识别和机器学习第 5.3 节。