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 .
