What is the differentiate of -2x/(1+x^2)^2?

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Answer 1 · 2018-02-26T09:50:10

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

We want to find the derivative of

#y=-(2x)/(1+x^2)^2#

Use the quotient rule, if #f(x)=g(x)/(h(x))#,

then #f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))^2#

For our problem

#g(x)=2x=>g'(x)=2#

#h(x)=(1+x^2)^2=>h'(x)=4x(1+x^2)#

Thus:

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

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

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

#=-(-6x^4-4x^2+2)/(1+x^2)^4#

#=(6x^4+4x^2-2)/(1+x^2)^4#

#=(2(3x^2-1)(x^2+1))/(1+x^2)^4#

#=(2(3x^2-1))/(1+x^2)^3#