This is a continuation from this question here. To reiterate, here is the question:
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x,y) = \alpha x^2 + \beta xy+ \gamma y^2$ for some $\alpha , \beta , \gamma \in \mathbb{R}^2$.
(A) Compute $Df(x)$ for $x \in \mathbb{R}^2$
(B) Show $Df(x)(y) = Df(y)(x)$ for all $x,y \in \mathbb{R}^2$
(C) Show $Df(x)(x) = 2f(x)$
I now understand that (B) is asking to prove that the directional derivatives $D_yf(x) = D_xf(y)$. How does one go about proving this $\forall x,y \in \mathbb{R}^2$?