Question

Dy/dx of ln(x/x^²+1)

Answer 1

`(dy)/(dx)\ \ln\left((x)/(x^2+1)\right)=(1)/((x)/(x^2+1))\cdot(x'(x^2+1)-x(x^2+1)')/((x^2+1)^2)\\\\=(x^2+1)/(x)\cdot(1(x^2+1)-x(2x))/((x^2+1)^2)=(1)/(x)\cdot(x^2+1-2x^2)/(x^2+1)=(1-x^2)/(x^3+x)\\\\Used:\\\\(\ln(x))'=(1)/(x)\\\\\left[f(g(x))\right]'=f'(g(x))\cdot g'(x)\\\\\left((f(x))/(g(x))\right)'=(f'(x)g(x)-f(x)g'(x))/([g(x)]^2)`