Exponential family
What is important to note, and what characterizes all exponential family variants, is that the parameter(s) and the observation variable(s) must factorize (can be separated into products each of which involves only one type of variable), either directly or within either part (the base or exponent) of an exponentiation operation. Generally, this means that all of the factors constituting the density or mass function must be of one of the following forms:
Exponential function
For the real exponential function is a constant and the derivative is zero because formula_3 for positive and the real exponential functions are monotonically increasing (as depicted for and ), because the derivative is greater than zero for all arguments, and for they are monotonically decreasing (as depicted for ), because the derivative is less than zero for all arguments.