If [tex]f(x)=12x-19[/tex] (whose domain is all real [tex]x[/tex]), then the derivative is
[tex]f'(x):=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h[/tex]
[tex]f'(x)=\displaystyle\lim_{h\to0}\frac{12(x+h)-19-(12x-19)}h=\lim_{h\to0}\frac{12h}h=\lim_{h\to0}12=12[/tex]
and similarly the domain of [tex]f'(x)[/tex] is all real [tex]x[/tex].
Please clarify your question if this is not the right interpretation of [tex]f(x)[/tex].
