Proof of The Derivative of a Vector-Valued Function as a Limit of Difference Quotients in Norm

Throughout, the arithmetic and order of $\mathbb{R}$ are used through The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix), Rules of Arithmetic and Order in an Ordered Field (iii) and The Quotient 1/x, Order Rules for Sums, and Nesting of Closed Intervals in ℝ (i); norms and distances in $\mathbb{R}^{n}$ through Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (iii); and the metric spaces $(\mathbb{R}^{n},d_{E})$ and $(\mathbb{R},d_{\mathbb{R}})$ through $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i) and The Absolute Value Metric on the Real Numbers (i).

For $s\in I$ with $s\neq t$ put

$$z(s)=\frac{\gamma(s)-\gamma(t)}{s-t}-c,\qquad\text{so that}\qquad z(s)_{k}=\frac{\gamma_{k}(s)-\gamma_{k}(t)}{s-t}-c_{k}\quad(k\in[q]),$$

the operations on $\mathbb{R}^{q}$ being coordinatewise. The whole proof rests on comparing the norm of $z(s)$ with its components: by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv),

$$|z(s)_{k}|\le\lVert z(s)\rVert\le\sum_{j=1}^{q}|z(s)_{j}|\qquad(k\in[q]).$$

The condition is necessary. Suppose that $\gamma$ is differentiable at $t$ with $\gamma'(t)=c$. By Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), each component $\gamma_{k}$ is differentiable at $t$ and $c_{k}=\gamma_{k}'(t)$. Let $\varepsilon>0$ be given. For each $k\in[q]$, $c_{k}=\gamma_{k}'(t)$ is a derivative of $\gamma_{k}$ at $t$ (Derivative of a Real Function on an Interval (ii)); choose $\delta_{k}>0$ by Derivative of a Real Function on an Interval (i) for $\gamma_{k}$ with tolerance $\varepsilon/(q+1)$, and then set $\delta=\min(1,\delta_{1},\dots,\delta_{q})$. For $s\in I$ with $0<|s-t|<\delta$ every component satisfies $|z(s)_{k}|<\varepsilon/(q+1)$, so the right-hand inequality of the display gives

$$\lVert z(s)\rVert\le\sum_{j=1}^{q}|z(s)_{j}|\le\frac{q\,\varepsilon}{q+1}<\varepsilon .$$

The condition is sufficient. Suppose that the condition holds, and let $k\in[q]$. Given $\varepsilon>0$, the $\delta>0$ that the condition provides for $\varepsilon$ gives, by the left-hand inequality of the display,

$$|z(s)_{k}|\le\lVert z(s)\rVert<\varepsilon\qquad(s\in I,\ 0<|s-t|<\delta).$$

So $c_{k}$ is a derivative of $\gamma_{k}$ at $t$ in the sense of Derivative of a Real Function on an Interval (i), and $\gamma_{k}$ is differentiable at $t$ with $\gamma_{k}'(t)=c_{k}$ (Derivative of a Real Function on an Interval (ii)). As this holds for every $k$, Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i) shows that $\gamma$ is differentiable at $t$ and that $\gamma'(t)$ has components $c_{k}$, that is, $\gamma'(t)=c$.