For a function $f(x)$ let $f'(x)$ denote its derivative.
By the definition of derivative, $\displaystyle f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}.$
By substituting $f$ with $f'$ in the above equation (the same formula should hold for any function and so we can do this), we get
$\displaystyle f''(x) = \lim_{h \to 0} \dfrac{f'(x+h) - f'(x)}{h}$
$\implies \displaystyle f''(x) = \lim_{h \to 0} \dfrac{\displaystyle \lim_{h \to 0} \dfrac{f(x+h + h) - f(x+h)}{h} - \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}}{h}$
$\implies \displaystyle f''(x) = \lim_{h \to 0} \dfrac{f(x+2h) - f(x+h) -{f(x+h) + f(x)}}{h^2}$
$\implies \displaystyle f''(x) = \lim_{h \to 0} \dfrac{f(x+2h) - 2f(x+h) + f(x)}{h^2}$
At $x = x_0-h$
$\displaystyle f''(x) = \lim_{h \to 0} \dfrac{f(x_0+h) - 2f(x_0) + f(x_0-h)}{h^2}$
Option $D.$
Ref: http://en.wikipedia.org/wiki/Second_derivative