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.