设 $f(x)$ 在 $x=x_0$ 处可导, 则 $\left.\lim _{\Delta x \rightarrow 0} \frac{f\left(x_0+\Delta x\right)-f\left(x_0\right)}{2 \Delta x}=\right.$
A. $\frac{1}{2} f^{\prime}\left(x_0\right)$
B. $2 f^{\prime}\left(-x_0\right)$
C. $f^{\prime}\left(x_0\right)$
D. $2 f^{\prime}\left(x_0\right)$