How do you differentiate #f(x)= x/(x^3-4x )# using the quotient rule?

1 Answer
Jim G.
Aug 3, 2016

#(-2x^3)/(x^3-4x)^2#

Explanation:

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

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

#color(red)(|bar(ul(color(white)(a/a)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(a/a)color(white)(a/a)|)))......(A)#
#"--------------------------------------------------------------"#
#g(x)=xrArrg'(x)=1#

#h(x)=x^3-4xrArrh'(x)=3x^2-4#
#"----------------------------------------------------------"#
Substitute these values into (A)

#f'(x)=((x^3-4x).1-x(3x^2-4))/(x^3-4x)^2#

#=(x^3-4x-3x^3+4x)/(x^3-4x)^2=(-2x^3)/(x^3-4x)^2#

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