exactvalue = N[7^(1/5),  3000];

p = N[1, 3000];  
    g[x_]  :=  x - (x^5 - 7)/(x^2);
For[i = 1,  
      i ≤ 14,  i++,  
{p = g[p],  
        y  =  Abs[p  -  exactvalue],  
Print[i],  
        Print[1.0*p],  Print[Log[y]/Log[10. ]],  
        Print[ ],  
}
]

g'[x]

1.0*g'[exactvalue]

p = N[1, 3000];   g[x_]  :=  
      x - (x^5 - 7)/(5*x^4);
For[i = 1,  
      i ≤ 14,  i++,  
{p = g[p],  
        y  =  Abs[p  -  exactvalue],  
Print[i],  
        Print[1.0*p],  Print[Log[y]/Log[10. ]],  
        Print[ ],  
}
]

g'[x]

1.0*g'[exactvalue]

p = N[1, 3000];   g[x_]  :=  
      x - (x^5 - 7)/12;
For[i = 1,  
      i ≤ 20,  i++,  
{p = g[p],  
        y  =  Abs[p  -  exactvalue],  
Print[i],  
        Print[1.0*p],  Print[Log[y]/Log[10. ]],  
        Print[ ],  
}
]

g'[x]

1.0*g'[exactvalue]

p = N[1, 3000];   g[x_]  :=  
      x - (x^5 - 7)/25;
For[i = 1,  
      i ≤ 20,  i++,  
{p = g[p],  
        y  =  Abs[p  -  exactvalue],  
Print[i],  
        Print[1.0*p],  Print[Log[y]/Log[10. ]],  
        Print[ ],  
}
]

g'[x]

1.0*g'[exactvalue]