Let $f=f(x_1,\dots,x_n)$ be a scalar function defined on some open subset $U\subset\mathbb{R}^n$. Given an unit vector $v=(v_1,\dots,v_n)$ and a point $x_0\in U$, the directional derivative of $f$ at $x_0$ in the direction of the vector $v$ is defined as the limit (provided it exists)$$D_v f(x_0):=\lim_{h\to 0}\frac{f(x_0+hv)-f(x_0)}{h}.$$
I recently founded another formula for directional derivative as the limit $$\lim_{h\to 0}\frac{f(x_0+\frac{h}{2}v)-f(x_0-\frac{h}{2}v)}{h}.$$ Are the two definitions equivalent?
Yes. To see this, first note that since this doesn't depend on whether the function is single-variable or multivariate, for convenience consider the former.
Thus, we have the limit of $$\frac{f(x+\epsilon)-f(x-\epsilon)}{2\epsilon},$$ with $\epsilon \ne 0.$ This can be written as $$\frac{f(x+\epsilon)-f(x)+f(x)-f(x-\epsilon)}{2\epsilon}=\frac12 \left(\frac{f(x+\epsilon)-f(x)}{\epsilon}+\frac{f(x)-f(x-\epsilon)}{\epsilon}\right)=\frac12 \left(\frac{f(x+\epsilon)-f(x)}{\epsilon}+\frac{f(y+\epsilon)-f(y)}{\epsilon}\right)=\frac12 \left(2\frac{f(z+\epsilon)-f(z)}{\epsilon}\right)=\frac{f(z+\epsilon)-f(z)}{\epsilon},$$ where I have set $x-\epsilon=y$ in the last displayed equation. Clearly the last displayed expression is the usual difference-quotient.