Why does the wrong derivative work better?

Question

Python

Hello! Write a multilayer perceptron with backpropagation according to this tutorial
It uses the logistic function and its derivative

def nonlin(self, x, deriv=False):
        if (deriv == True):
            return x * (1 - x)
        return 1 / (1 + np.exp(-x))

and it turns out that the derivative is x * (1 - x), but judging by Google, the derivative of the logistic function is df'(x) = f(x) * (1 - f(x)), and not df'(x)= x * (1 - x). However, when I correct the calculation of the derivative to the correct version in the code, the prediction accuracy of the neural network drops significantly.
Why?

Answer 1

Because when calculating the derivative, the nonlin input is not the input of this layer, but the previously calculated output.