def softmax(X):
  exps = np.exp(X - X.max())
  return exps / np.sum(exps)

def softmax_prime(X, output):
  result = np.zeros(X.shape)

  for i in range(len(output)):
    for j in range(len(X)):
      if i == j:
        result = output[i] * (1 - X[i])
      else:
        result = -output[i] * X[j]

  return result