just been very confused about the definition of these two derivatives the past few days.

$\lim\limits\_{h \to 0} \frac{f(a+h)-f(a)}{h}$

and

$\lim\limits\_{h \to 0} \frac{f(x+h)-f(x)}{h}$

When do i use them? and what are the differences between these two "definitions" and how come they're called definitions and not formulas?