class Softmax:
  # ...

  def backprop(self, d_L_d_out):
    '''
    Performs a backward pass of the softmax layer.
    Returns the loss gradient for this layer's inputs.
    - d_L_d_out is the loss gradient for this layer's outputs.
    '''
    # We know only 1 element of d_L_d_out will be nonzero
    for i, gradient in enumerate(d_L_d_out):
      if gradient == 0:
        continue

      # e^totals
      t_exp = np.exp(self.last_totals)

      # Sum of all e^totals
      S = np.sum(t_exp)

      # Gradients of out[i] against totals
      d_out_d_t = -t_exp[i] * t_exp / (S ** 2)
      d_out_d_t[i] = t_exp[i] * (S - t_exp[i]) / (S ** 2)

      # Gradients of totals against weights/biases/input
      d_t_d_w = self.last_input
      d_t_d_b = 1
      d_t_d_inputs = self.weights

      # Gradients of loss against totals
      d_L_d_t = gradient * d_out_d_t

      # Gradients of loss against weights/biases/input
      d_L_d_w = d_t_d_w[np.newaxis].T @ d_L_d_t[np.newaxis]
      d_L_d_b = d_L_d_t * d_t_d_b
      d_L_d_inputs = d_t_d_inputs @ d_L_d_t

      # ... to be continued