How to get the loss function derivative

Question

I am following a lecture on logistic regression using gradient descent and I have an issuer understanding a short-path for a derivative :

Let be :

• $z=w_1x_1+w_2x_2+b$
• $a=\sigma(z)$
• and the loss function $L(a,y)=-y(\log(a)+(1-y)\log(1-a))$, which I know have a name but I can't remember it it.

In order to have $\frac{\delta\mathcal L(a,y)}{\delta z}$ I am able to compute $\frac{\delta\mathcal L(a,y)}{\delta a}$. But where do we get $\frac{\delta a}{\delta z}=a(1-a)$ ?

I can only have $\frac{\delta a}{\delta z}=\frac{\delta\sigma(w_1x_1+w_2x_2+b)}{\delta z}$

Answer 1

$$\sigma(z)=\frac{1}{1+\exp(-z)}$$

\begin{align}\frac{\partial a}{\partial z}&=\frac{\exp(-z)}{(1+\exp(-z))^2}\\ &=\frac{1}{1+\exp(-z)}\frac{\exp(-z)}{1+\exp(-z)}\\ &=\frac{1}{1+\exp(-z)}\left(1- \frac{1}{1+\exp(-z)}\right)\\ &=\sigma(z)(1-\sigma(z))\\ &=a(1-a)\end{align}

Answer 2

If $\sigma(x) = \frac{1}{1+\exp{(-x)}}$ then $\sigma^\prime(x) = \frac{-1}{[1+\exp(-x)]^2}\cdot(-\exp(-x)) = \frac{\exp(-x)}{[1+\exp(-x)]^2}$ (by the quotient rule)

And on the other hand, $\sigma(x)\cdot [1-\sigma(x)]$ is

$$\begin{align*} \frac{1}{1+\exp{(-x)}} \cdot \left[1-\frac{1}{1+\exp{(-x)}}\right] &= \frac{1}{1+\exp{(-x)}} \cdot \left[\frac{1+\exp{(-x)}}{1+\exp{(-x)}} - \frac{1}{1+\exp{(-x)}}\right]\\&=\frac{\exp(-x)}{\left[1+\exp(-x)\right]^2}\end{align*}$$

which is the same thing. So, $\sigma^\prime(x) = \sigma(x) \cdot [1-\sigma(x)]$.