= and := can be a little tricky when trying to write Newton.

I won't do the whole problem for you, because this might be homework, but I will show you some hints that should help you get started.

In[1]:= aold = 0; bold = 0; cold = 0;

In[2]:= f1 = -.036((.9-a)^1.852)+0.032((1.05+a)^1.852)+0.038((.85+a)^1.852)+0.693((.25+a+ b)^1.852);

In[3]:= df1 = D[f1, a]

Out[3]= 0.066672(0.9-a)^0.852+0.070376(0.85+a)^0.852+0.059264(1.05+a)^0.852+1.28344(0.25+a+b)^0.852

In[4]:= anew = aold - (f1/df1 /. {a->aold, b->bold, c->cold})

Out[4]= -0.15003

Study this. If it is too complicated then try a simpler f1 and study that. See if you can relate each part of this to the definition of Newton from the textbook. Do this until you think you see it is working correctly for the first step with the first variable. When you think you have that working then think how to add a second and third function and variable to this and just do the first step of Newton.

When you think you have that correct the try doing the second step manually and see if that is correct.

After you have several manual steps done then you might think about how to make the steps repeat without having to do this manually.

Always check very carefully to make certain that I have not made a mistake in this.

Hopefully these will be enough hints to get you started. If you get stuck then post again showing what you have done and how much progress you have made and what part you cannot see how to do.