Question

请问f(x)=(e^x-1)(e^2x-2)...(e^nx-n),f(0)的导数是多少?

Answer 1

f(x)=(e^x-1)(e^2x-2)...(e^nx-n),g(x)=(e^2x-2)...(e^nx-n),f(x)=(e^x-1)g(x),

f'(x)=(e^x-1)'g(x)+(e^x-1)g'(x)=e^xg(x)+(e^x-1)g'(x),f'(0)=g(0)=(1-2)(1-3)-----(1-n)=(-1)^(n-1)1×2×----×(n-1)=(-)^(n-1)(n-1)!