Why do so many functions used in data science have derivatives of the form f(x)*(1-f(x))?
Sigmoid function is a partial case of softmax, when the number of classes $K=2$. That's why the similarity of their derivatives shouldn't surprise you.


  Why do so many functions used in data science have derivatives of the
  form f(x)*(1-f(x))?


If you consider the following differential equation

$y' = y \cdot (1-y)$

you will find the general solution in the form

$y(x) = \frac{e^x}{c + e^x} = \frac{1}{1 + c e^{-x}}$

So, in some sense, there are not that many functions with this property: they are "very like" sigmoid function.