Show the derivative of an activation function

Question

I am learning about neural networks and am using the sigmoid activation function

$$q(z)=\frac{1}{1+e^{-z}}.$$ The problem is that I need to use its derivative $q^{\prime}(z)$. Would anyone have any hints as to how I would go about calculating this?

Answer 1

Hint: $$\left(\frac 1f\right)^{\prime}=-\frac{f^{\prime}}{f^2}$$ when $f$ is non zero and differentiable. This means $$q^{\prime}(z)=\left(\frac{1}{1+e^{-z}}\right)^{\prime}=-\frac{(1+e^{-z})^{\prime}}{(1+e^{-z})^2}$$