how the cross-entropy speed up backpropagation on the hidden layer?

Question

I am learning the http://neuralnetworksanddeeplearning.com/chap3.html.

It says the cross-entropy cost function can speed up the network, because the δ^'(z) canceled on the last layer.

the partial derivative for the last layer L:

∂(C)/∂(w) = a^L-1(a^L-y).

there is no δ^'(z).

But I want to know whether cross-entropy speed up the hidden-layer, so I calculate the partial derivative on the L-1 layer:

∂(C)/∂(w)

= (a^L-y) * w^L * a^L-1(1-a^L-1) * a^L-2

= (a^L-y) * w^L * δ'(z^L-1) * a^L-2

It seems it not speed up on the L-1 layer, because δ'(x) still exists. I can be close to zero the make the partial derivative close to zero, and make the learning slowly.

Could someone tell me the wrong point I made ? Thanks.