Another completely different method and arguably more intuitive is to use asymptotic expansions. Proof by asymptotic expansion For any differentiable function $f$ such that $f'$ is differentiable at $x$ we have: $f(x+h) \in f(x) + f'(x) h + \frac{1}{2}f''(x) h^2 + o(h^2)$ as $h \to 0$ This uses...