设函数f(x)在x=2的某领域内可微,且f'(x)=e^f(x),f(2)=1,求f'''(2)
问题描述：
设函数f(x)在x=2的某领域内可微,且f'(x)=e^f(x),f(2)=1,求f'''(2)
答
f'(x)=e^f(x)①
当x=2时,f(x)=1,那么f'(2)=e^f(2)=e
①式两边同时对x进行求导,得：f''(x)=e^f(x)*f'(x)=e^f(x)*e^f(x)=e^[2f(x)] ②
将x=2,f(2)=1代入,得：f''(2)=e^[2f(2)]=e^2
②式两边同时对x进行求导,得：f'''(x)=e^[2f(x)]*2f'(x)=2e^[2f(x)]*e^f(x)=2e^[3f(x)]
将x=2,f(2)=1代入,得：f'''(2)=2e^[3f(2)]=2e^3