How do you differentiate #f(x) = (e^(4-x))/(e^(1-x)-1)# using the quotient rule?

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Answer 1 · 2017-04-20T23:34:56

#f'(x) = e^(4-x) / (e^(1-x)-1)^2#

#f(x)=u/v, f'(x)=(u'v-v'u)/v^2#

So, we need #u'# and #v'#:

#u' = (e^(4-x))' = e^(4-x) * (4-x)' = -e^(4-x)#

#v' = (e^(1-x) - 1)' = e^(1-x) * (1-x)' = -e^(1-x)#

#f'(x) = (-e^(4-x) * (e^(1-x) - 1) + e^(1-x) * e^(4-x)) / (e^(1-x)-1)^2#

#f'(x) = (-e^(5-2x) + e^(4-x) + e^(5-2x)) / (e^(1-x)-1)^2#

#f'(x) = e^(4-x) / (e^(1-x)-1)^2#