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

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Answer 1 · 2018-04-09T18:34:53

#(de^x/(xe^x-4))/(dx)=-e^x(e^x+4)/(xe^x-4)^2#.

#d(g(x)/(h(x)))/(dx)=(g'(x)*h(x)-g(x)*h'(x))/[h(x)]^2#

Here

#g(x)=e^x#,

#g'(x)=e^x#,

#h(x)=xe^x-4#, and

#h'(x)=e^x+xe^x# by the product rule

so

#(de^x/(xe^x-4))/(dx)=(e^x(xe^x-4)-e^x(e^x+xe^x))/(xe^x-4)^2#

#=-e^x(e^x+4)/(xe^x-4)^2#.