Question

设f(√x)=sinx 则f`[f(x)]=? 求高人指点

则后边的第一个f后边那点好像看不怎么清楚，那是求导哦!高人指点下
我看书上的答案是2sinx^2Xcos(sinx^2)^2，我觉得答案不对啊，你们参谋参谋吧

Answer 1

令a=√x
则x=a²
所以f(a)=sin(a²)
f(x)=sin(x²)
f[f(x)]=sin[sin²(x²)]
所以f'[f(x)]=cos[sin²(x²)]*[sin²(x²)]'
=cos[sin²(x²)]*2sin(x²)*[sin(x²)]'
=cos[sin²(x²)]*2sin(x²)*cos(x²)*(x²)'
=2xcos[sin²(x²)]*2sin(x²)*cos(x²)

Answer 2

f(√x)=sinx =sin(√x)^2=>f(x)=sin(x^2)
f'(f(x))=(sin((sinx^2)^2))'
=cos((sinx^2)^2))*(((sinx^2)^2))'
=cos((sinx^2)^2))*2sin(x^2)(sin(x^2))'
=cos((sinx^2)^2))*2sin(x^2)cos(x^2)*2x