Vrftjkry

I am trying to prove this equation (from the backpropagation equations in AI).

$$frac{partial C}{partial b_j^l} = delta_j^l$$

C is the cost function: $C = frac{1}{2}||y - a^L||^2$

Where the output of layer l and neuron j is express like so $a^l_j=σ(∑_kw^l_{jk}a^{l−1}_k+b^l_j)$

I am suppose to use this assertion to do the demonstration: $delta_j^L = frac{partial C}{partial z_j^L}$

So far, here is what I have tried:

$$frac{partial C}{partial z_j^L} = sum_k frac{partial C}{partial z_j^L} frac{partial b_j^l}{partial b_j^L} $$ (I am using the chain rule to have a sum)

<=> $$frac{partial C}{partial z_j^L} = sum_k frac{partial C}{partial b_j^l} frac{partial b_j^l}{partial z_j^L} $$

So I guess I have to prove, that $frac{partial b_j^l}{partial z_j^L}$ equals 1.

But I don't have any ideas how to prove it.

Thanks for your help

N.B
I am following this course => http://neuralnetworksanddeeplearning.com/chap2.html where the first two equations of the BackPropagation equations are already proved, and the 2 others should be proved the same way (using the chain rule)