Confusion about differential of multivariable functions. I have been given a function $F: \mathbb{R²} \to \mathbb{R³}$
and another function $\beta: J \to \mathbb{R³}: t \mapsto F(a + tx)$
where $x = v_1(1,0) + v_2(0,1) = (v_1,v_2)$ and $a$ is fixed. we can assume that all given functions are differentiable on their domain.
I'm asked to find $\beta'(0)$ (this should be equal to $D_1F(a)v_1 + D_2F(a)v_2$)
My attempt:
Let $R(t) = a+tx$
Then $\beta'(0) = D\beta(0) = D(F\circ R)(0) = DF(R(0)) \circ DR(0)$
and then I'm stuck.

$R(t)=(R_{1}(t),R_{2}(t)):=(a_{1}+tx_{1},a_{2}+tx_{2})$, and $\beta=F\circ R$, so $\beta'(0)=\dfrac{\partial F}{\partial x}(R(0))R_{1}'(0)+\dfrac{\partial F}{\partial y}(R(0))R_{2}'(0)$. Note that $R_{1}'(0)=x_{1}$ and $R_{2}'(0)=x_{2}$.
Here $\dfrac{\partial F}{\partial x}=D_{1}F$ and similar to $D_{2}F$.
