1,f(x)=(πtan sec x)^6
f'(x)=[6(πtan sec x)^5]×[πsec^2(sec x)]×[secx tanx]
=6π^6(tan secx)^5×(sec secx)^2×secx×tanx
令g(x)=πtan sec x
f(x)=[g(x)]^6
由复合函数的求导规则
f'(x)=6[g(x)]^5×g'(x)
g'(x)=πtan'(sec x)×sec'x
={π[sec (sec x)]^2}×(secx tanx).
2,f(x)=arcsin(sinx+1/2)
f'(x)={1/√[1-(sinx+1/2)^2]}×cos x.
令g(x)=sin x+1/2
这里要注意f(x)的定义域,-1≤sin x+1/2≤1
即-1≤sin x≤1/2.
定义域为：[-π/2+2kπ,π/6+2kπ]∪[5π/6+2kπ,2π+2kπ].
f'(x)=1/√{1-[g(x)]^2}×g'(x)
g'(x)=cos x.