How do you use the quotient rule to differentiate #y= (x^2+4x)/(x+2)^2#?

1 Answer
Jim G.
Jun 3, 2016

#8/(x+2)^3#

Explanation:

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

#f(x)=(g(x))/(h(x))" then "f'(x)=(h(x).g'(x)-g(x)h'(x))/(h(x))^2#
#"---------------------------------------------------------------------"#
#g(x)=x^2+4xrArrg'(x)=2x+4#

#h(x)=(x+2)^2rArrh'(x)=2(x+2)#
#"-----------------------------------------------------------------------------"#
Substitute these values into f'(x)

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

#=((x+2)[(x+2)(2x+4)-2(x^2+4x)])/(x+2)^4#

#=(cancel((x+2))[2x^2+4x+4x+8-2x^2-8x])/(cancel((x+2))(x+2)^3#

#=8/(x+2)^3#

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