Differentiation

Abstract

Let f be defined on (a, b), and choose c ∈ (a, b). We say that f is differentiable at c if

$$\displaystyle{\lim _{x\rightarrow c}\frac{f(x) - f(c)} {x - c} }$$

exists as a real, i.e., exists and is not $\pm \infty$. If it exists, we denote this limit f′(c) or $\dfrac{df} {dx}(c)$, and we say that f′(c) is the derivative of f at c. If f is differentiable at c for all a < c < b, we say that f is differentiable on (a, b) or, if it is clear from the context, differentiable. In this case, the derivative $f': (a,b) \rightarrow \mathbf{R}$ is a function defined on all of (a, b).