Single Layer Perceptron Backpropagation: How to compute affect of the net value on the output?

Question

Assuming a single perceptron (see figure), I have found two versions of how to use backpropagation to update the weights. The perceptron is split in two, so we see the weighted sum on the left (the output of this is net) and then the sigmoid function phi on the right (the output of this is out).

So for the backpropagation portion, we compute $\frac{\delta cost}{\delta w}=\frac{\delta cost}{\delta out}\times \frac{\delta out}{\delta net} \times \frac{\delta net}{\delta w}​$, to find how the weight affects the error.

For the portion $\frac{\delta out}{\delta net}$ I have seen three different versions for how to compute the value:

1. $\frac{\delta out}{\delta net}$ = $\phi'(out) = \phi(out) \times (1 - \phi(out))$
2. $\frac{\delta out}{\delta net}$ = $\phi'(net)= \phi(net) \times (1 - \phi(net))$
3. $\frac{\delta out}{\delta net}$ = $out \times (1 - out)$

Can somebody explain to me which one is correct and why? Or is there one which should be preferred?

Answer 1

1. is always wrong. Most implementations then "silently" correct this error, even if the docstrings have this wrong variant, in the actual use it is done correctly.

2. and 3. are the same, considering that $out=\phi(net)$. 3. is more economic if the activation is a solution of an autonomous ODE like with the logistic function. 2. is more generally true, especially if the activation function gets modified/perturbed by some linear function to avoid random behavior in the flat parts of the sigmoid at large values.