summarize the pertinent information obtained by applying the graphing strategy of f(x)=2x/(1-x^2)?

Question

need to know how to do first derivative

Answer 1

f'(x) = ( (1-x^2)2-2x(-2x)) /(1-x^2)^2
=(2-2x^2+4x^2)/(1-x^2)^2
=(2+2x^2)/(1-x^2)^2
now
f'(x)=0 then
2+2x^2=0, but this equation does not have solutions, so
there are no critical points,
and f'(x)>0, so f(x) is always increasing.
However, it has vertical asymptotes at x=1,-1

Answer 2

Quotient rule:

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

The derivative is never zero, so the function has no local maxima or minima. However, it has vertical asymptotes at x=1,-1.