By definition,
f′(2)=limx→2f(x)−f(2)x−2=limx→211−2x−11−4x−2=limx→211−2x+13x−2=limx→2(1x−23+(1−2x)3(1−2x))=limx→2(1x−2−2(x−2)3(1−2x))=limx→2−23(1−2x)=−23⋅(−3)=29.
Alternatively,
f′(2)=limh→0f(2+h)−f(2)h=limh→011−4−2h−11−4h=limh→01−3−2h+13h=limh→0(1h3−3−2h3(−3−2h))=limh→0(1h−2h3(−3−2h))=limh→023(3+2h)=29.
