How do you differentiate #f(x) = (e^x-x)/(2-e^x)# using the quotient rule?

1 Answer
Jim G.
Nov 3, 2016

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

Explanation:

differentiate using the #color(blue)"quotient rule"#

#"If" f(x)=(g(x))/(h(x))" then"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'x)/[h(x)]^2)color(white)(2/2)|)))#

here #g(x)=e^x-xrArrg'(x)=e^x-1#

and #h(x)=2-e^xrArrh'(x)=-e^x#

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

distribute the numerator and collect like terms.

#rArrf'(x)=(2e^x-2-e^(2x)+e^x-(-e^(2x)+xe^x))/(2-e^x)^2#

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

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