How do you differentiate #f(x) =x/(x^3-e^(x)-1)# using the quotient rule?

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Answer 1 · 2017-07-28T17:58:30

The derivative is #=((xe^x-1-2x^3-e^x))/(x^3-e^x-1)^2#

#(u/v)'=(u'v-uv')/(v^2)#

We have

#f(x)=(x)/(x^3-e^x-1)#

Here,

#u(x)=x#, #=># #u'(x)=1#

#v(x)=x^3-e^x-1#, #=>#, #v'(x)=3x^2-e^x#

Therefore,

#f'(x)=(1*(x^3-e^x-1)-x(3x^2-e^x))/(x^3-e^x-1)^2#

#=((x^3-e^x-1-3x^3+xe^x))/(x^3-e^x-1)^2#

#=((xe^x-1-2x^3-e^x))/(x^3-e^x-1)^2#