Question

关于y=sin（x/2）的求导,我不知道我的推导哪里错了

Answer 1

错在第一步.大意失荆州.

f(x)=sin(x/2)

f(x+⊿x)=sin(x/2+⊿x/2)

⊿y=sin(x/2+⊿x/2)-sin(x/2)=2cos(x/2+⊿x/4)sin(⊿x/4)

⊿y/⊿x=2cos(x/2+⊿x/4)sin(⊿x/4)/⊿x=cos(x/2+⊿x/4)sin(⊿x/4)/[2(⊿x/4)]

所以（⊿x→0）lim⊿y/⊿x=lim[cos(x/2+⊿x/4)]•1/2*[ sin(⊿x/4)/(⊿x/4)]=1/2*cos(x/2)