[tex]f(x)=(1+x)^6\implies f'(x)=6(1+x)^5\implies f'(0)=6[/tex]
At [tex]x=0[/tex], you get [tex]f(0)=(1+0)^6=1[/tex], so the tangent line to [tex]f[/tex] at this point is
[tex]y-1=6(x-0)\implies y=6x+1[/tex]
which suggests [tex]f(0.1)\approx6\times0.1+1=1.6[/tex]
