You can do this two ways.
a)
Quotient Rule:
d/(dx)[(f(x))/(g(x))] = (g(x)(df(x))/(dx) - f(x)(dg(x))/(dx))/((g(x))^2ddx[f(x)g(x)]=g(x)df(x)dx−f(x)dg(x)dx(g(x))2
d/(dx)[x/(1+x^2)] = [(1+x^2)(1) - (x)(2x)]/(1+x^2)^2ddx[x1+x2]=(1+x2)(1)−(x)(2x)(1+x2)2
= (1+x^2-2x^2)/(1+x^2)^2=1+x2−2x2(1+x2)2
= (1-x^2)/[(1+x^2)^2]=1−x2(1+x2)2
b)
Product Rule + Chain Rule:
d/(dx)[f(x)*g(x)] = f(x)(dg(x))/(dx) + g(x)(df(x))/(dx)ddx[f(x)⋅g(x)]=f(x)dg(x)dx+g(x)df(x)dx
d/(dx)[x*(1/(1+x^2))] = d/(dx)[x*(1+x^2)^(-1)]ddx[x⋅(11+x2)]=ddx[x⋅(1+x2)−1]
= [x][-(1+x^2)^(-2)*(2x)] + [1/(1+x^2)][1]=[x][−(1+x2)−2⋅(2x)]+[11+x2][1]
= -(2x^2)/(1+x^2)^2 + 1/(1+x^2)=−2x2(1+x2)2+11+x2
= [-2x^2 + 1 + x^2]/(1+x^2)^2=−2x2+1+x2(1+x2)2
= [1 - x^2]/(1+x^2)^2=1−x2(1+x2)2