How do you find the 1000th derivative of #y=xe^-x#?

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Answer 1 · 2018-04-01T17:56:01

#y^((1000))=-1000e^-x+xe^-x#

Try to determine a general pattern for the derivative of the function by taking the first few derivatives.

#y'=e^-x-xe^-x#
#y''=-e^-x-e^-x+xe^-x=-2e^-x+xe^-x#
#y'''=2e^-x+e^-x-xe^-x=3e^-x-xe^-x#
#y^((4))=-3e^-x-e^-x+xe^-x=-4e^-x+xe^-x#

So, we have a prety good pattern going.

The #nth# derivative, where #n# is even ( #n=2, 4,...#), follows the pattern

#y^((n))=-n e^-x+xe^-x#

We want #y^((1000)),# the above pattern is enough for us, no need to try to determine the odd pattern as well.

#y^((1000))=-1000e^-x+xe^-x#