分段函数求导x≠0时f(x)=(1-cosx)/xx=0时f(x)=0求f'(0)=?求详解

Question

分段函数求导 x≠0时 f(x)=(1-cosx)/x x=0时f(x)=0 求f'(0)=?求详解

Answer 1

因为lim(x→0）(1-cosx)/x
=lim(x→0）sinx
=0
所以函数是连续函数,可导.
f'(x)=[sinx*x-(1-cosx)]/x^2
=(xsinx-1+cosx)/x^2
对导数求极限=(sinx+xcosx-sinx)/2x=cosx/2=1/2.