In Milnor's Topology From the Differentiable Viewpoint, the derivative of a smooth map $f: U\to V$ is defined as $$ \mathrm{d}f_x: \mathbb{R}^k \to \mathbb{R}^l $$ $$ h\mapsto \lim_{t\to0} \frac{f(x+th)-f(x)}{t} $$ On the fifth page he remarks
Clearly $\mathrm{d}f_x (h)$ is a linear function of $h$.
What does he mean here? This would require that $\mathrm{d} f_x (h_1+h_2)= \mathrm{d} f_x(h_1)+\mathrm{d} f_x(h_2)$, but this is does not seem to the case in general.
I think Milnor means what he says: $df_x(h)$ is a linear function of $h$ (not of $f$, as @Jp McCarthy suggested). I think "clearly" is overstating the case, however. In fact, the usual definition of what it means to say that $f$ is differentiable at $\boldsymbol x$ is equivalent to the statement that the limit $$ \lim_{t\to0} \frac{f(x+th)-f(x)}{t} $$ exists and depends linearly on $h$. Milnor's claim that this is the case relies on a theorem that a function of class $C^1$ (meaning that all first partial derivatives exist and are continuous) is differentiable at every point. This can be proved, for example, by using Taylor's theorem, or more directly by applying the fundamental theorem of calculus to the function $t\mapsto f(x+th)$.