# Backpropagation with log likelihood cost function and softmax activation

In the online book on neural networks by Michael Nielsen, in chapter 3, he introduces a new cost function called as log-likelihood function defined as below
$$C = -ln(a_y^L)$$ Suppose we have 10 output neurons, when back propagating the error, only the gradient w.r.t. $$y^{th}$$ output neuron is non-zero and all others are zero. Is that right?

If so, how is the below equation (81) true? $$\frac{\partial C}{\partial b_j^L} = a_j^L - y_j$$ I'm getting the expression as $$\frac{\partial C}{\partial b_j^L} = y_j (a_j^L - 1)$$