As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \beta as \mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(i-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right), where we had defined the logistic (sigmoid) function p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, and p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}). The parameters \hat{\beta} were defined using a minimization method like gradient descent or Newton-Raphson's method.
Now we replace x_i with the activation z_i^l for a given layer l and the outputs as y_i=a_i^l=f(z_i^l) , with z_i^l now being a function of the weights w_{ij}^l and biases b_i^l . We have then a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, with z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, where the superscript l-1 indicates that these are the outputs from layer l-1 . Our cost function at the final layer l=L is now \mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), where we have defined the targets t_i . The derivatives of the cost function with respect to the output a_i^L are then easily calculated and we get \frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. In case we use another activation function than the logistic one, we need to evaluate other derivatives.