One way of checking whether your code is correct or not is to try it on a simple example, can it find where Sin[x] is zero starting at x==1?
In[1]:= NewtonsMethodList[f_, {x_, x0_}, n_] := NestList[#-(Function[x, f][#]*
Derivative[1][Function[x, f]][#])/((Derivative[1][Function[x, f][#]])^2 -
Function[x, f][#]*Derivative[2][Function[x, f]][#]) &, x0, n];
NewtonsMethodList[Sin, {x, 1}, 6]
Out[2]= {1, 1, 1, 1, 1, 1, 1}
I did remove spaces in your "Newtons Method List" and "Nest List" because I assumed those were both supposed to be single word identifiers. I also was worried about the precedence of ^ versus your use of Derivative so I wrapped Derivative in () and moved the ^2 outside.
As you can see, your method isn't converging on zero like you would expect for Sin[x].
Here is a completely different way of trying to calculate your function.
In[3]:= Clear[f, fp, fpp];
f[x_] := Sin[x];
fp[x_] = D[f[x], x];
fpp[x_] = D[f[x], {x, 2}];
nm[pm1_] := pm1 - (f[pm1] fp[pm1])/(fp[pm1]^2 - fp[pm1] fpp[pm1]);
N[NestList[nm, 1, 8]]
Out[8]= {1., 0.391021, 0.0991114, 0.00866767, 0.0000742696, 5.51543*10^-9, 2.77556*10^-17, 0., 2.4259*10^-18}
That doesn't try to pass in your function as an argument or calculate your derivatives on the fly, but it does seem to converge on zero like I would expect. This very different style of coding isn't probably what you are expecting to turn in.
I'd suggest you don't try to solve your problem by typing in the whole thing and then wondering whether it is right or not. Instead I'd suggest you start much smaller with a simpler goal, like seeing if you can get NestList to correctly just find {f[x], f[f[x]], f[f[f[x]]]} for a really simple function f. See if you can figure out all the things wrong with doing that and finally get that working. Then work on the next harder problem, see if you can get NestList to correctly just find {f'[x], f'[f'[x]], f'[f'[f'[x]]]}. Once you have figured out how to do both those then you might be able to combine those to produce the result you are actually supposed to get.