:PROPERTIES: :ID: a42a8f1c-52b3-4d7e-9692-c91fbdeaec61 :mtime: 20231028012431 :ctime: 20231028012430 :END: #+title: differentiation (mathematics) #+filetags: :public:project: * Derivative The derivative is a function \[D : (a \to b) \to a \to (a \to b)\] * Single-variable derivative \[\frac{df}{dx} = \frac{f(x + \eps) - f(x)}{\eps}\] * Differentiability A function that has a derivative is called *differentiable*. Not all functions are differentiable. Consider $f(x) = |x|$. Then, $D(f)(0)$ is undefined. ** Single-variable differentiability A function is said to be *differentiable* iff 1) $f$ is continuous 2) The limit $\frac{f(x-\eps)-f(x)}{\eps}$ exists 3) The left-hand limit and the right-hand limit agree. * Common Derivative Rules ** Power Rule \[D(x \mapsto x^{a})(c) = y \mapsto ac^{a-1}y\] ** Chain Rule \[D(g \circ f)(x) = D(g)(f(x)) \circ D(f)(x)\]