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:
$= color(blue)(e^(-x)(x-2))$

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