Two things are required to solve this problem. The chain rule, and knowledge of the derivative of #ln(x)#. The chain rule states that,
#"d"/("d"x) "f"("g"(x)) = "g"'(x)"f"'("g"(x))#.
And the derivative of #ln(x)# is #1/x#.
Then,
#"d"/("d"x) ln(x+sqrt(x^2+1)) = ("d"/("d"x) (x+sqrt(x^2+1))) * 1/(x+sqrt(x^2+1))#.
Then the derivative of #x+sqrt(x^2+1)# needs to be computed.
#"d"/("d"x) (x+(x^2+1)^(1/2)) = 1 + 1/2 (x^2+1)^(-1/2) *2x#,
#"d"/("d"x) (x+(x^2+1)^(1/2)) = 1 + x/sqrt(x^2+1)#,
#"d"/("d"x) (x+(x^2+1)^(1/2)) = (sqrt(x^2+1) + x)/(sqrt(x^2+1))#.
Then, substituting,
#"d"/("d"x) ln(x+sqrt(x^2+1)) = (sqrt(x^2+1) + x)/(sqrt(x^2+1)) * 1/(x+sqrt(x^2+1))#.
The factor of #x+sqrt(x^2+1)# cancels.
#"d"/("d"x) ln(x+sqrt(x^2+1)) = 1/sqrt(x^2+1)#.
Incidentally, the function #ln(x+sqrt(x^2+1))# is the inverse hyperbolic sine function. You can read more about hyperbolic functions and their derivatives here .
