Definition 4.4.1. Let \(I\subset \mathbb {R}\) be an open interval and let \(f:\, I \longrightarrow \mathbb {R}\). Suppose that \(f\) is differentiabl at all points of \(I\). If \(c\in I\) is such that \(f'\) is itself differentiable at \(c\), then \(f\) is said to posses a second order derivative at \(c\) denoted by \(f''(c)\). Thus \[f''(c) = \lim _{x\rightarrow c} \, \dfrac {f'(x) - f'(c)}{x - c}\]

We generalize this to give the definition of an \(n^{\text {th}} - \) order derivative.

Definition 4.4.2. Let \(I\subset \mathbb {R}\) be an open interval and \(C\in I\). If \(f:\, I\longrightarrow \mathbb {R}\) possesses an \((n - 1)^{\text {th}}\) order derivative at all points of \(I\) and \(n\geq 2\). If the \((n - 1)^{\text {th}}\) order derivative \(f^{n-1}\) is differentiable at \(c\), then its derivative is called the \(n^{\text {th}}\) order derivative of \(f\) at \(c\) denoted \(f^n(c)\). Thus \[f^n(c) = \lim _{x\rightarrow c} \, \dfrac {f^{(n - 1)}(x) - f^{(n-1)}(c)}{x - c}\]

Definition 4.4.3. If \(f\) possesses on \(n^{\text {th}}\) order derivatives at all points of \(I\), we say that \(f\) is \(n\) times differentiable on \(I\).