I thought about this problem a bit more and we are really asking to show that

$\displaystyle \operatorname{\frac{d}{dt}}(f\circ F)(0) = \operatorname{\frac{d}{dt}}(g\circ F)(0)$

where$\displaystyle F: t \mapsto p + tv$. That is,

$\displaystyle \displaystyle\lim_{h \rightarrow 0} \frac{f\circ F(h) - f\circ F(0)}{h} = \displaystyle\lim_{h \rightarrow 0} \frac{g\circ F(h) - g\circ F(0)}{h}$

But since we care about the limit we can simply choose $\displaystyle h$ small enough so that $\displaystyle p+hv$ is contained in some ball contained in $\displaystyle W$ and hence these two limits are equal