Definition 3

We say that a vector function $\mathbf{f}$ defined on an interval $I \subset \mathbf{R}$ is differentiable (resp. right differentiable, left differentiable) on $I$ if it is differentiable (resp. right differentiable, left differentiable) at each point of $I$; the function $x \mapsto \mathbf{f}'(x)$ (resp. $x \mapsto \mathbf{f}'_d(x)$, $x \mapsto \mathbf{f}'_g(x)$) defined on $I$, is called the derived function, or (by abuse of language) the derivative (resp. right derivative, left derivative) of $\mathbf{f}$, and is denoted by $\mathbf{f}'$ or $\mathrm{Df}$ or $d\mathbf{f}/dx$ (resp. $\mathbf{f}'_d, \mathbf{f}'_g$).