Proof of Derivatives along Lines in ℝⁿ: the Line t ↦ x+th, and Partial Derivatives as Derivatives along Coordinate Axes

The derivative of the line. In (i), let $t\in J$ and let $\varepsilon>0$ be real; take $\delta=1$. For $s\in J$ with $0<|s-t|<1$ we have $s-t\neq0$ and, by Lines, Scalar Multiples, Basis Directions, Small Balls and Matrix Rows in ℝⁿ (iv), $\gamma(s)-\gamma(t)=(x+sh)-(x+th)=(s-t)h$. The quotient is the scalar multiple by $(s-t)^{-1}$, with $s-t\neq0$; by the law $s(tf)=(st)f$ of Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iii) and by Lines, Scalar Multiples, Basis Directions, Small Balls and Matrix Rows in ℝⁿ (ii),

$$\frac{\gamma(s)-\gamma(t)}{s-t}-h=\bigl((s-t)^{-1}(s-t)\bigr)h-h=h-h=0,$$

using Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii) in the last step, and $\lVert0\rVert=0<\varepsilon$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iii). By The Derivative of a Vector-Valued Function as a Limit of Difference Quotients in Norm (i), with $c=h$, $\gamma$ is differentiable at $t$ with $\gamma'(t)=h$.

Partial derivatives give derivatives along an axis. In (ii), put $a=x+se_{i}$, a point of $U$. Since $\partial_{i}F(a)$ exists and equals $L$, the number $L$ is a partial derivative of $F$ with respect to the $i$-th variable at $a$ (Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii)). Let $\varepsilon>0$ be real. By Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (i) there is a real $\delta>0$ such that every real $t$ with $0<|t|<\delta$ satisfies $a+te_{i}\in U$ and

$$\Bigl|\frac{F(a+te_{i})-F(a)}{t}-L\Bigr|<\varepsilon .$$

Let $u\in J$ with $0<|u-s|<\delta$, and take $t=u-s$. By Lines, Scalar Multiples, Basis Directions, Small Balls and Matrix Rows in ℝⁿ (iii), $a+te_{i}=x+(s+(u-s))e_{i}=x+ue_{i}$, so $F(a+te_{i})=\varphi(u)$ and $F(a)=\varphi(s)$, and the display reads

$$\Bigl|\frac{\varphi(u)-\varphi(s)}{u-s}-L\Bigr|<\varepsilon .$$

Thus $L$ is a derivative of $\varphi$ at $s$ in the sense of Derivative of a Real Function on an Interval (i), and by Derivative of a Real Function on an Interval (ii) $\varphi$ is differentiable at $s$ with $\varphi'(s)=L$.

Derivatives along an axis give partial derivatives. In (iii), $(-r,r)$ is an interval: if $u,w\in(-r,r)$ and $y$ is real with $u\le y\le w$, then $-r<u\le y\le w<r$, so $y\in(-r,r)$ by Intervals of the Real Line (ii). It contains the distinct points $0$ and $r/2$, as $-r<0<r/2<r$ by Rules of Arithmetic and Order in an Ordered Field (ix). Moreover $\varphi(0)=F(x+0e_{i})=F(x)$ by Lines, Scalar Multiples, Basis Directions, Small Balls and Matrix Rows in ℝⁿ (i). Let $\varepsilon>0$ be real. First, Derivative of a Real Function on an Interval (i), applied to the derivative $L$ of $\varphi$ at $0$, gives a real $\delta_{0}>0$ such that every $t\in(-r,r)$ with $0<|t|<\delta_{0}$ satisfies

$$\Bigl|\frac{\varphi(t)-\varphi(0)}{t}-L\Bigr|<\varepsilon .$$

Next, Order Rules for Products, Halves, Minima and Absolute Values in ℝ (ix) gives a real $\delta>0$ with $\delta\le\delta_{0}$ and $\delta\le r$. Let $t$ be real with $0<|t|<\delta$. Then $|t|<r$, so $-r<t<r$ by Order Rules for Products, Halves, Minima and Absolute Values in ℝ (x), that is, $t\in(-r,r)$; hence $x+te_{i}\in U$, $\varphi(t)=F(x+te_{i})$, and $|t|<\delta_{0}$ gives

$$\Bigl|\frac{F(x+te_{i})-F(x)}{t}-L\Bigr|<\varepsilon .$$

So $L$ is a partial derivative of $F$ with respect to the $i$-th variable at $x$ in the sense of Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (i); by Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii), $\partial_{i}F(x)$ exists and equals $L$.