定義より、
f'(x) = lim[h→0] ( f(x+h)-f(x) )/h

ここで、
f(x+h)-f(x)
= ( a(x+h\()^{2}\)+b(x+h)+c ) - ( a\(x^{2}\)+bx+c )
= a( (x+h\()^{2}\)-\(x^{2}\) ) + b( (x+h)-x )
= 2axh + a\(h^{2}\) + bh

よって、
f'(x)
= lim[h→0] ( 2axh + a\(h^{2}\) + bh )/h
= lim[h→0] ( 2ax + b + ah )
= 2ax+b