The sigmoid function, σ(x) = 1 / (1 + e^-x), is a cornerstone in logistic regression and neural networks, mapping any real value to a range between 0 and 1. This transformation allows outputs to be interpreted as probabilities, such as predicting a student's pass/fail status based on study hours. However, the function's true origin lies in the mathematical constant 'e', approximately 2.71828, which emerges naturally from continuous compounding interest.

Consider a bank offering a 100% annual interest rate: with yearly compounding, $100 grows to $200. If compounded semi-annually, the amount becomes $225, and as compounding frequency increases infinitely, the growth factor approaches 'e'. This limit defines exponential growth, and the sigmoid function is essentially a smoothed version of this exponential behavior, making it ideal for activation in neural networks.

Understanding this derivation clarifies why the sigmoid is used and highlights its role in mitigating vanishing gradients, a common challenge in deep learning.