I'm having a lot of problems figuring out this question. If someone could at least point me in the right direction that would be great.
"Prove that for every positive integer n , (d^n / dx^n)[xe^-x] = (-1)^n e^-x(x-n)"
I'm having a lot of problems figuring out this question. If someone could at least point me in the right direction that would be great.
"Prove that for every positive integer n , (d^n / dx^n)[xe^-x] = (-1)^n e^-x(x-n)"
Do this by induction.
$\displaystyle \frac{{d^k }}{{dx^k }}\left[ {xe^{ - x} } \right] = \left( { - 1} \right)^k e^{ - x} \left( {x - k} \right)$
$\displaystyle \begin{array}{lr} {\frac{{d^{k + 1} }}
{{dx^{k + 1} }}\left[ {\left( { - 1} \right)^k e^{ - x} \left( {x - k} \right)} \right]} & { = \left( { - 1} \right)^k \left[ { - e^{ - x} \left( {x - k} \right) + e^{ - x} } \right]} \\
{} & {=\left( { - 1} \right)^{k + 1} \left( {e^{ - x} } \right)\left[ {x - \left( {k + 1} \right)} \right]} \\
\end{array} $