Derivative Formal Definition

Definitions

First Principles

${\lim }_{x\to 0}\frac{f(x+h)-f(x)}{h}$

a, b

${\lim }_{x\to a}\frac{f(x)-f(a)}{x-a}$

One-sided Derivatives

Mostly used for Piecewise Functions

Left Sided Derivatives

${\lim }_{h\to 0^{-}}\frac{f(a+h)-f(a)}{h}$

Right Sided

${\lim }_{h\to 0^{+}}\frac{f(a+h)-f(a)}{h}$