----- > [!proposition] Proposition. ([[existence of derivative guarantees existence of all directional derivatives]]) > Let $\Omega$ be an [[open set]] in $\mathbb{R}^{m}$; let $f:\mathbb{R} ^{m} \to \mathbb{R}^{n}$ be differentiable at $a \in \Omega$. Then $f'(\vec a; \vec u)$ exists for all $\vec u$ with $f'(\vec a; \vec u) = Df(\vec a) \vec u$. > [!proof]- Proof. ([[existence of derivative guarantees existence of all directional derivatives]]) > Let $B = Df(\vec a)$. Set $\vec h = t \vec u$ in the definition of the derivative as in the above proof, where $t \neq 0$. We have $0 = \lim_{ t \to 0 } \frac{f(\vec a+t \vec u ) - f(\vec a)-Bt\vec u}{\vert t \vec u \vert}. $ Assuming $t$ approaches 0 through postive values, we multiply both sides of the above by $\vert \vec u \vert$ to concludes that $\frac{f(a+t\vec u)-f(\vec a)}{t} - B\vec u \to 0.$ else multiply instead by $- \vert \vec u \vert$; in either case the desired result is obtained. \ BUT NOTICE! $\frac{f(a+t\vec u)-f(\vec a)}{t}$ is just $f'(\vec a; \vec u)$! And so we see that $f'(\vec a; \vec u)= B\vec u=$ $Df(\vec a) \vec u$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```