function defined by the average rate of change

function defined by the average rate of change

Given a differentiable function $f(x)$, let $g(x)$ be defined by $g(x) =
\begin{cases} (f(x)-f(a))/(x-a) &\mbox{if } x \neq a \\ f'(a) & \mbox{if }
x = a. \end{cases}$
Suppose also f(x) is twice differentiable, then I guess that
$g'(a)=f''(a)/2$ since $$g'(a)= \lim \frac{g(x)-g(a)}{x-a}=\lim
\frac{f(x)-f(a)-f'(a)(x-a)}{(x-a)^2}$$ and applying L'Hospital's rule
twice. But how can I prove this without using L'Hospital's rule?