How do you differentiate #f(x)= ( 13x^2+ 3x ) / ( 4e^x + 2 ) # using the quotient rule?

1 Answer
Jim G.
Nov 24, 2016

#f'(x)=(e^x(92x-52x^2+12)+52x+6)/(4e^x+2)^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)=13x^2+3xrArrg'(x)=26x+3#

and #h(x)=4e^x+2rArrh'(x)=4e^x#

#rArrf'(x)=((4e^x+2)(26x+3)-(13x^2+3x).4e^x)/(4e^x+2)^2#

#=(104xe^x+12e^x+52x+6-52x^2e^x-12xe^x)/(4e^x+2)^2#

#=(92xe^x-52x^2e^x+12e^x+52x+6)/(4e^x+2)^2#

#=(e^x(92x-52x^2+12)+52x+6)/(4e^x+2)^2#

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