Is this correct?
Assume that $(D_{v}f)(x)>0$ for some point $x\in \mathbb{R}^{n}$ and some vector $v\in \mathbb{R}^{n}$. Then,
\begin{align} (D_{-v}f)(x) &= \lim_{h \to 0}\frac{f(x-hv)-f(x)}{h}\\ &= - \lim_{h \to 0}\frac{f(x-hv)-f(x)}{-h}\\ &= -(D_{v}f)(x)<0 \end{align} I know that maybe this question is trivial but I am not sure if I am understanding well.