求导2arctan(x+sqrt(1+x^2))?1/(1+x^2)

Question

求导2arctan(x+sqrt(1+x^2))?
1/(1+x^2)

Answer 1

=2*1/(1+(x+sqrt(1+x^2))^2)*(x+sqrt(1+x^2))'
=2/[1+x^2+2xsqrt(1+x^2)+1+x^2]*(1+1/2(1+x^2)^(-1/2)*2x)
=2[1+x(1+x^2)^(-1/2)]/[2+2x^2+2xsqrt(1+x^2)]
=[1+x(1+x^2)^(-1/2)]/[1+x^2+xsqrt(1+x^2)]
=[1+xsqrt(1+x^2)/(1+x^2)]/[1+x^2+xsqrt(1+x^2)]
={[(1+x^2+xsqrt(1+x^2)]/(1+x^2)}*{1/[1+x^2+xsqrt(1+x^2)]}
分子和分母约掉
=1/(1+x^2)