Problem : $(\mathbb{R}^n,\|\ \|)$ has a smooth and strictly convex norm. When $f(x)=\|x\|$, then find a directional derivative of a function $f$, i.e. $\frac{d}{dt}f(x+tv)$ for $\|x\|=1$.
Refer : [1] On intrinsic geometry of surface in normed spaces - Burago and Ivanv
[2] Convex analysis - Rockafellar
When $S$ is $\|\ \|$-unit sphere, then $N(x),\ x\in S$ is out unit normal vector. Then we have a supporting function $h_x$ s.t. $h_x(V)=\frac{\langle N,V\rangle}{\langle N,x\rangle}$.
Hence we have $$\frac{d}{dt}f(x+tv) =h_x(v) $$