We need to know that #d/dxlog(x)=1/x#. The chain rule then tells us that #d/dxlog(f(x))=1/f(x)f'(x)#.
Then:
#d/dxlog(x+sqrt(x^2+a^2))=1/(x+sqrt(x^2+a^2))d/dx(x+sqrt(x^2+a^2))#
So now all we need to do is find the derivative of #x+sqrt(x^2+a^2)#. The derivative of #x# is #1# and we find the derivative of #sqrt(x^2+a^2)# by doing the chain rule on #(x^2+a^2)^(1/2)#.
Then the derivative of the original function is:
#=1/(x+sqrt(x^2+a^2))(1+1/2(x^2+a^2)^(-1/2)d/dx(x^2+a^2))#
And the derivative of #x^2+a^2# is #2x#:
#=1/(x+sqrt(x^2+a^2))(1+1/(2sqrt(x^2+a^2))(2x))#
#=1/(x+sqrt(x^2+a^2))(1+x/(sqrt(x^2+a^2)))#
#=1/(x+sqrt(x^2+a^2))((sqrt(x^2+a^2)+x)/sqrt(x^2+a^2))#
#=1/sqrt(x^2+a^2)#
