3512.sagews

#f(x)= (x^2)-ln(1+x)#
#fp(x)=derivative(f(x))# 
#phi1(x)=x-(f(x)/fp(x))#

def f(x): 
    return ((x^2)-ln(1+x)) 
def fp(x):
    return (2*x-1/(x+1))
def phi1(x):
    return (x-(f(x)/fp(x))) 
def newton1(x_start,eps):
    x=x_start 
    x_old=x+10*eps 
    while(abs(x-x_old)>eps): 
         x_old=x 
         x=phi1(x) 
         return(x)

newton1(0.1,1.e-7)

-0.0203092279291762

fp(1)

3/2

plot(f(x),-0.9,5)