What is the derivative of #e^-x(1-x)#?

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Answer 1 · 2015-07-30T00:11:16

You can always say:

#g(x) = e^(-x)#
#h(x) = (1-x)#

Thus you can use the product rule, which is:

#d/(dx)[g(x)h(x)] = g(x)h'(x) + h(x)g'(x)#

So:

#(dy)/(dx) = (e^(-x))(-1) + (1-x)(-e^(-x))#

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

Distribute operations:
#= -e^(-x) - e^(-x) + xe^(-x)#

Combine like terms:
#= -2e^(-x) + xe^(-x)#

Factor out like terms:
#= color(blue)(e^(-x)(x-2))#