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.

You're almost done.
The differential of an affine function $x \mapsto Ax + b$ where $A$ is a linear map and $b$ is a vector is equal to the linear map $A$. Indeed:
$$\lim_{h\to 0}\frac{\|(Ah + b) - (A\cdot 0 + b) - Ah\|}{\|h\|} = \lim_{h\to 0}\frac{\|Ah + b - Ah - b\|}{\|h\|} = 0$$
Hence, we have $DR(0) = x\cdot$, the operator of multiplication by $x$.
$DF(R(0))$ is just $DF(a)$, since $R(0) = a$.
So, $D\beta(0) = DF(a) \,\circ x\cdot$.
