Derivative of sigmoid?

Question

I'm creating a neural network using the backpropagation technique for learning. I understand we need to find the derivative of the activation function used. I'm using the standard sigmoid function f(x) = 1 / (1 + e^(-x)) and I've seen that its derivative is dy/dx = f(x)' = f(x) * (1 - f(x)) This may be a daft question, but does this mean that we have to pass x through the sigmoid function twice during the equation, so it would expand to dy/dx = f(x)' = 1 / (1 + e^(-x)) * (1 - (1 / (1 + e^(-x)))) or is it simply a matter of taking the already calculated output of `f(x)`, which is the output of the neuron, and replace that value for `f(x)`?

Answer 1

You don't need to expand a function to use its result.

The fact that its result is being used multiple places in the derivative doesn't change that.

e.g. when we write that two things are equal, they are literally exactly equivalent. So use what f(x) works out to or the expression that explains how it gets worked out --- wherever, and as many times as you want.