HELP! derivative definition

[lsp2010]

Use limit as ?x->0 f(a+?x)-f(a)/?x to find

f prime (2) where f(x)=x^2 -3x + 1

Here's what I've done. Correct my mistakes please.

lim ?x->0 ((c+?x)^2-3(c+?x)+1)/?x
lim ?x->0 ((2+?x)^2-3(2+?x)+1)/?x
lim ?x->0 (-1+?x+?x^2)/?x
lim ?x->0 -1+?x^2

and now i am stuck.

 

[lsp2010]

What do I do next? Just plug 0 in for deltax? Or have I made mistakes?

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ x^{2}-3x+1, \ f'(x) \ = \ \lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)

\(\displaystyle =\lim_{h\to0}\frac{(x+h)^{2}-3(x+h)+1-x^{2}+3x-1}{h}=\lim_{h\to0}\frac{x^{2}+2xh+h^{2}-3x-3h+1-x^{2}+3x-1}{h}\)

\(\displaystyle =\lim_{h\to0}\frac{2xh+h^{2}-3h}{h} \ = \ \lim_{h\to0} \frac{h(2x+h-3)}{h} \ = \ \lim_{h\to0}2x+h-3 \ = \ 2x-3\)

\(\displaystyle f'(x) \ = \ 2x-3, \ f'(2) \ = \ 1\)