Definition 1

Let $f$ be a vector function defined on an interval $I \subset \mathbf{R}$ which does not reduce to a single point. We say that $f$ is differentiable at a point $x_0 \in I$ if

exists (in the vector space where $f$ takes its values); the value of this limit is called the first derivative (or simply the derivative) of $f$ at the point $x_0$, and it is denoted by $f'(x_0)$ or $Df(x_0)$.

If $f$ is differentiable at the point $x_0$, so is the restriction of $f$ to any interval $J \subset I$ which does not reduce to a single point and such that $x_0 \in J$; and the derivative of this restriction is equal to $f'(x_0)$. Conversely, let $J$ be an interval contained in $I$ and containing a neighbourhood of $x_0$ relative to $I$; if the restriction of $f$ to $J$ admits a derivative at the point $x_0$, then so does $f$.

We summarise these properties by saying that the concept of derivative is a local concept.