How do you differentiate #f(x)=(-4x)/(x^2-1)# using the quotient rule?

2 Answers
Jim G.
May 20, 2018

#f'(x)=(4x^2+4)/(x^2-1)^2#

Explanation:

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

#f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=-4xrArrg'(x)=-4#

#h(x)=x^2-1rArrh'(x)=2x#

#rArrf'(x)=(-4(x^2-1)+8x^2)/(x^2-1)^2#

#color(white)(rArrf'(x))=(4x^2+4)/(x^2-1)^2#

James
May 20, 2018

#y'=[4x^2+4]/(x^2-1)^2#

Explanation:

show below

#color(blue)[y=(-4x)/(x^2-1)]#

#color(red)[y=[f(x)]/[g(x)]]#

The quotient rule #color (red)[y'=(g(x)f'(x)-f(x)g'(x))/(g(x))]#

#y'=[(x^2-1)*(-4)-(-4x)(2x)]/(x^2-1)^2#

#y'=[-4x^2+4+8x^2]/(x^2-1)^2=[4x^2+4]/(x^2-1)^2#

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