Here you need to prove an identity. So, start with one side of the equation and arrive at something. Then start messing with the other side and try to reach that same something.
If you have already proved the product rule, this little trick may come in handy:
[tex]\frac{1}{x}=\frac{1}{x}^2x[\tex]
Yes, they wouldn't ask you to use Newton's Method if they only wanted the trivial root. Try to start with a first approximation that is near the root. I'd try 1.