Differences between Newton's raphson and bisection method
Answers
Newton's method is great for speed, but it does require that you know the derivative, and I have yet to encounter a real-world application where this was available. That is not to say that they don't occur. But I have not been so lucky. Another problem with Newton's method is instability. If you hit a place where the function is close to flat, it may send your next iteration out beyond Pluto. And in fact, there is no guarantee of convergence. You can find it getting caught in a loop.
Bisection Method. Guaranteed convergence, provided you can straddle the root at the start. Easily understood, easily programmed, easily performed, slow as blazes. Never sends your iteration off into the wild blue yonder. But still slow as blazes. This is your fallback method when all else fails.