Calculus I - Simple Difference Quotient Question The problem is to calculate the difference quotient of $f(x) = \sqrt{x^2 +2x+1}$. But $\sqrt{x^2+2x+1}= \sqrt{x+1}^2 = x+1$ so can I just take the difference quotient of $x+1$? If not, how do I simplify $\frac{\sqrt{x^2+2xh+h^2+2x+2h+1} - \sqrt{x^2+2x+1}}{h}$ ? 

Not observing the fact that $\sqrt{x^2+2x+1}=x+1$, that lonely $h$ will factor out of the denominator in your other form if you multiply the entire mess by 
$$\frac{\sqrt{x^2+2xh+h^2+2x+2h+1} + \sqrt{x^2+2x+1}}{\sqrt{x^2+2xh+h^2+2x+2h+1} + \sqrt{x^2+2x+1}}.$$
Things will suddenly start to cancel in the numerator significantly. You will then factor an $h$ completely out of what remains in the numerator, and with that cancel the $h$ out of the denominator. Proceed cautiously, this is error prone.
Observing your simplification fact is a good idea, much more straightforward, and mathematically acceptable. The results either way will be identical.
