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f(x)= (x^2 -1)/(x^2 + 1)

Let f(x) = u/v such that:

u = (x^2 -1)  ==> u' = 2x

v= x^2 + 1  ==>   v' = 2x

==> f'(x)= (u'v-uv')/v^2

              = (2x(x^2 + 1) - (x^2 -1)2x ]/(x^2 +1)^2

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

              = 4x/(x^2 + 1)

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

 

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