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

In the setting of Multivariable Calculus: Notation, let $x\in\mathbb{R}^{n}$, and let $J$ be an interval containing two distinct points.

(i) The derivative of the line. Let $h\in\mathbb{R}^{n}$. If $\gamma:J\to\mathbb{R}^{n}$ satisfies $\gamma(t)=x+th$ for every $t\in J$, then $\gamma$ is differentiable at every $t\in J$, with $\gamma'(t)=h$.

(ii) Partial derivatives give derivatives along an axis. Let $F:U\to\mathbb{R}$, let $i\in[n]$, and let $\varphi:J\to\mathbb{R}$ satisfy $x+te_{i}\in U$ and $\varphi(t)=F(x+te_{i})$ for every $t\in J$. Let $s\in J$ and $L\in\mathbb{R}$. If the partial derivative $\partial_{i}F(x+se_{i})$ exists and equals $L$, then $\varphi$ is differentiable at $s$ with $\varphi'(s)=L$.

(iii) Derivatives along an axis give partial derivatives. Let $F:U\to\mathbb{R}$ with $x\in U$, let $i\in[n]$ and $L\in\mathbb{R}$, let $r>0$ be real, and let $\varphi:(-r,r)\to\mathbb{R}$, on the open interval $(-r,r)$, an interval containing two distinct points, satisfy $x+te_{i}\in U$ and $\varphi(t)=F(x+te_{i})$ for every $t\in(-r,r)$. If $\varphi$ is differentiable at $0$ with $\varphi'(0)=L$, then $\partial_{i}F(x)$ exists and equals $L$.