Single-variable calculus from first principles
Taylor's real payoff in ML is linearisation: replacing a stubborn nonlinear function with its tangent line near a point of interest. Over a small range the linear approximation is nearly exact, and linear things are far easier to analyse, compute, and reason about.
The sigmoid σ(x) = 1/(1 + e⁻ˣ) is the familiar squashing nonlinearity. Near x = 0 it passes through ½ with slope ¼, so its linear approximation is:
A flat paper street map treats the round Earth as a plane near one city. Over a few kilometres the curvature is too tiny to matter, so the flat sheet is accurate enough to navigate by, even though the planet is really a sphere. Linearisation does the same to a function: near a point it swaps the true curve for the tangent line f(x) ≈ f(0) + f′(0)·x, exact enough locally and far easier to work with.