Derivative of a Vector-Valued or Matrix-Valued Map on an Interval

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $q\in\mathbb{N}$, let $I$ be an interval containing two distinct points, as in Single-Variable Calculus on an Interval (ii), and let $t\in I$.

(i) Let $x:I\to\mathbb{R}^{q}$. For $k\in[q]$, $x_{k}:I\to\mathbb{R}$ is the component map $u\mapsto x(u)_{k}$ of Component Maps and Entry Maps (i), with the components of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii). $x$ is differentiable at $t$ if $x_{k}$ is differentiable at $t$ for every $k\in[q]$. The derivative of $x$ at $t$ is then the point $x'(t)\in\mathbb{R}^{q}$ with components $x'(t)_{k}=x_{k}'(t)$ for $k\in[q]$: each $x_{k}'(t)$ is the unique derivative of $x_{k}$ at $t$ by Derivative of a Real Function on an Interval (ii), the map $[q]\to\mathbb{R}$, $k\mapsto x_{k}'(t)$, is formed by Predicative Formulas for the Derivative Condition and for Powers (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v) and is a point of $\mathbb{R}^{q}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii), and a point of $\mathbb{R}^{q}$ is determined by its components by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii).

(ii) Let $p\in\mathbb{N}$ and $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$. For $i\in[p]$ and $j\in[q]$, $A_{ij}:I\to\mathbb{R}$ is the entry map $u\mapsto A(u)_{ij}$ of Component Maps and Entry Maps (ii), with the entries of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii). $A$ is differentiable at $t$ if $A_{ij}$ is differentiable at $t$ for all $i\in[p]$ and $j\in[q]$. The derivative of $A$ at $t$ is then the matrix $A'(t)\in\mathcal{M}_{p\times q}(\mathbb{R})$ with entries $A'(t)_{ij}=A_{ij}'(t)$ for $i\in[p]$ and $j\in[q]$: each $A_{ij}'(t)$ is the unique derivative of $A_{ij}$ at $t$ by Derivative of a Real Function on an Interval (ii), the map $[p]\times[q]\to\mathbb{R}$, $(i,j)\mapsto A_{ij}'(t)$, is formed by Predicative Formulas for the Derivative Condition and for Powers (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v) and is a matrix by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), and a matrix is determined by its entries by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii).

(iii) For a subset $S\subseteq I$, a map $x:I\to\mathbb{R}^{q}$, respectively, for $p\in\mathbb{N}$, a map $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$, is differentiable on $S$ if it is differentiable at every point of $S$ in the sense of (i), respectively (ii); it is differentiable if it is differentiable on $I$.

(iv) If $x:I\to\mathbb{R}^{q}$ is differentiable, its derivative is the map $x':I\to\mathbb{R}^{q}$, $u\mapsto x'(u)$; if $p\in\mathbb{N}$ and $A:I\to\mathcal{M}_{p\times q}(\mathbb{R})$ is differentiable, its derivative is the map $A':I\to\mathcal{M}_{p\times q}(\mathbb{R})$, $u\mapsto A'(u)$. These maps are formed by Predicative Formulas for the Derivative Condition and for Powers (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v), the values $x'(u)$ and $A'(u)$ being those of (i) and (ii) with $u$ in place of $t$.