Question

Find the 600th derivative of
f(x) = xe−x.

Answer 1

do first 4 derivatives and spot the pattern

\[f'(x) = e^{-x} - x e^{-x}\]
\[f''(x) = -e^{-x} - f'(x) = - 2e^{-x} +x e^{-x} \]
\[f'''(x) = f'(x)+2e^{-x} = 3 e^{-x} - x e^{-x}\]
\[f^{4} (x) = -4 e^{-x} + x e^{-x}\]
...
see the pattern

Answer 2

ok and how do you set up the equation from the pattern seen?

Answer 3

so it just changes signs and adds one to first coefficient

Answer 4

yeah basically

formally it would would be:
\[f^n (x) = (-1)^n (x e^{-x} - n e^{-x})\]

Answer 5

ah i see! thank you