I tried to prove it, but I would appreciate if someone could check my answer. I am just starting to learn real analysis on my own Thank you for helping. 🙂
Theorem
Let $f\colon \Bbb R^2 \to \Bbb R$ have continuous first partial derivatives at each point.
Then $f$ has directional derivatives in all directions at each point.
Proof
Let $\vec p$ be any non-zero vector in $\Bbb R^2$.
Choose $r>0$.
By the mean value proposition, we see that if $t$ is any number such that $|t|\lVert \vec p \rVert < r$, then there are $n$ points $z_1,\ldots,z_n\in\Bbb R^2$ such that
$$f(x+t\vec p)-f(x)=\sum_{i=1}^n t p_i \frac{\partial f}{\partial x_i}(z_i)$$
and $\lVert z_i – x \rVert < |t|\lVert p \rVert$ for each $i\in\{1,\ldots,n\}$.
Then we can write
$$\frac{f(x+t\vec p)-f(x)} t =\sum_{i=1}^n p_i \frac{\partial f}{\partial x_i}(z_i)$$
Since $\frac{\partial f_i}{\partial x_i}$ is continuous for each $i$,
$$\lim_{t\to 0} \frac{f(x+t\vec p)-f(x)} t = \lim_{t\to 0}\sum_{i=1}^n p_i \frac{\partial f}{\partial x_i}(z_i)$$ exists, so $f$ has a derivative in the $\vec p$ direction at $x$.
Best Answer
I think you also forgot to normalize your vector $\,\vec p\,$ . Taking it from where you were, and assuming we already have $\,||\vec p||=1\,$ , we have by definition with $\,\bf x:=(a_1,a_2,...,a_n)\;$
$$D_{\vec p}f(\bf x):=\text{The directional derivative of $\,f\,$ at point $\,\bf x\,$ in the direction of $\,\vec p$}:=$$
$$:= \lim_{t\to 0}\frac{f(x+t\vec p)-f(x)}t\stackrel{\text{Chain Rule}}=\sum_{k=1}^n \vec p\cdot \frac{\partial f}{\partial x_k}(a_1,...,a_n)=\nabla f(\bf{x})\cdot\bf \vec p$$
So if the function is differentiable at any point (say, when the partial derivatives exist and are continuous at any point), its directional derivative in any direction exists and it's pretty easy to calculate by means of the gradient of $\,f\,$ .
Added: We want the directional derivative of $\;f\;,\;f:\Bbb R^n\to\Bbb R\;$ at the point $\,x:=(x_1,...,x_n)\,$ and in the direction $\,p:=(p_1,...,p_n)\,$ (no little arrows, no nothing) .
We also assume $\,||p||:=\sqrt{\sum_{k=1}^n p_i^2}=1\;$ (note then that you will have to normalize the vector in which direction you want the derivative in case it is not normalized!).
Now we define a new function
$$g:\Bbb R\to\Bbb R^n\;,\;\;g(t):=x+tp=\left(x_1+tp_1\,,\,x_2+tp_2\,,\ldots,x_n+tp_n\right):=(g_1(t),...,g_n(t))$$
Note that $\;\forall\,t\;,\;\,g'(t)=p\implies g'(0)=p\,$ , and we also denote the derivative of some function $\,k\,$ by $\,Dk\,$ and by $\,D_uk\,$ the directional derivative of $\,k\,$ in the direction of $\,u\,$, for simplicity.
Thus, by definition we get that
$$D_pf(x):=\lim_{t\to 0}\frac{f(x+tp)-f(x)}t=\left.\frac d{dt}(f(x+tp))\right|_{t=0}=\left.D(f\circ g)(t)\right|_{t=0}\stackrel{\text{chain rule}}=$$
$$=Df(g(0))\cdot Dg(0)=\left(f'_{x_1}(g(0))\,,\,f'_{x_2}(g(0))\,,\ldots,f'_{x_n}(g(0))\right)\cdot(g'_1(0)\,,\,\ldots,g'_n(0))=$$
$$\left(f'_{x_1}(g(0))\,,\,f'_{x_2}(g(0))\,,\ldots,f'_{x_n}(g(0))\right)\cdot(p_1,...,p_n)=\nabla f(x)\cdot p$$
which is what we wrote above...:)
IMPORTANT: You can use the gradient as above to calculate the directional derivative only if you're sure the partial derivatives exist in some neighborhood of $\,x\,$ and are continuous there.