Answer:
[tex]\frac{d^{200}}{dx^{200}}(xe^{-x})=e^{-x}(x-200)[/tex]
Step-by-step explanation:
The given function is
[tex]f(x)=xe^{-x}[/tex]
Differential with respect to x.
[tex]f'(x)=\frac{d}{dx}(xe^{-x})[/tex]
[tex]f'(x)=x\frac{d}{dx}e^{-x}+e^{-x}\frac{d}{dx}x[/tex]
[tex]f'(x)=x(-e^{-x})+e^{-x}(1)[/tex]
[tex]f'(x)=-xe^{-x}+e^{-x}[/tex]
[tex]f'(x)=-e^{-x}(x-1)[/tex]
Differential with respect to x.
[tex]f''(x)=e^{-x}x-2e^{-x}[/tex]
[tex]f''(x)=e^{-x}(x-2)[/tex]
Differential with respect to x.
[tex]f'''(x)=-e^{-x}(x-3)[/tex]
Similarly, the nth derivative is
[tex]\frac{d^{n}}{dx^{n}}(xe^{-x})=(-1)^ne^{-x}(x-n)[/tex]
The 200th derivative of f(x) is
[tex]\frac{d^{200}}{dx^{200}}(xe^{-x})=e^{-x}(x-200)[/tex]
