Derivatives in terms of $z_j^L$

It is also easy to see that our previous equation can be written as

$$\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},$$

which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely

$$\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},$$

That is, the error $\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias.

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