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