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)
