Given that we have the sigmoid function:
$$\sigma(x) = \frac{1}{1 + e^{-x}}$$
What is its derivative?
It can be rewritten as
$$\sigma (x) = (1 + e^{-x})^{-1}$$
Say that we have:
$$m(x) = -x$$
$$t(m) = e^{m}$$
$$h(t) = 1 + t$$
$$g(h) = h^{-1}$$
Then, we have
$$\sigma(x) = g(h)$$
So:
$$\begin{aligned} \frac{d\sigma}{dx} & = \frac{dg}{dh}\cdot \frac{dh}{dt}\cdot \frac{dt}{dm}\cdot \frac{dm}{dx} \ &= (-h^{-2})\times …