Why does this limit define a linear map between the tangent spaces?

Question

Let $U \subset \mathbb{R}^n$, $V \subset \mathbb{R}^k$ be open, $f\colon U \to V$ a smooth map, $x \in U$. For $h \in \mathbb{R}^n$, let $$ d_x(h) := \lim_{t \to 0} \frac{f(x+ th) - f(x)}{t} $$ Now, in "Topology from the differential viewpoint", Milnor writes that clearly, $d_x$ is a linear function of $h$. Why is that so? In other words, how could one see that $d_x$ is simply the Jacobian matrix?

Answer 1

Define $c_x(t) = x + th$ so $t \mapsto f(c(t))$ is differentiable at zero with derivative (by Chain Rule) $d_x(h) = f'(x)c'(0)=f'(x)h...$ linear in $h.$