# Homework Help: Another Iffy Derivative

1. Jul 27, 2011

### 1MileCrash

1. The problem statement, all variables and given/known data

$f(x) = ln(x+\sqrt{x^2+1})$

2. Relevant equations

3. The attempt at a solution

First, I applied the chain rule.

$[\frac{1}{x+\sqrt{x^2+1}}]Dx[x+\sqrt{x^2+1}]$

Second, to find $Dx[x+\sqrt{x^2+1}]$, I broke it into two derivatives. Derivative of x is 1, so

$1 + Dx[\sqrt{x^2+1}]$

To find $Dx[\sqrt{x^2+1}]$, I applied the chain rule once more.

$[\frac{1}{2}][2x]\frac{1}{\sqrt{x^2+1}}$

I simplified this result to:

$\frac{x}{\sqrt{x^2+1}}$

$[\frac{1}{x+\sqrt{x^2+1}}][1+\frac{x}{\sqrt{x^2+1}}]$

The book gives a much cleaner answer of $\frac{1}{\sqrt{x^2+1}}$

Is my answer equivalent? If yes, how would I get to that? If no, what part of the calculus did I screw up?

WOW, Nevermind!

Last edited: Jul 27, 2011
2. Jul 27, 2011

### SammyS

Staff Emeritus
Do some algebra.

Find a common denominator for $\displaystyle 1+\frac{x}{\sqrt{x^2+1}}\,.$ & write as one fraction.