On your two options, I want to say that you have to take it on the former...except that the the LHS should be f(x) and not zero as posted...because you need the f'(x) in the Newton-Raphson method.
The second equation is not equivalent to the first equation.
But, question, is the (1 +x^2) in the numerator really not the same as the (1 +x) in the denominator?
Because if both were (1 +x^2), or both were (1 +x), then you can decompose the RHS of the first equation into two simpler fractions that might not make you mad to differentiate.
Thanks for the help! I think figured out how to rewrite the equation to make taking the derivative a little easier. Does anyone know of a program to put the original function and the derivative into in order to do the iterations? I have been trying to do them by a combination of pencil/paper and graphing calculator, but that is getting really complicated.