x0=8.3;
x1=8.3;
x2=8.6;
x3=8.6;
h0=x1-x0;
h1=x2-x1;
h2=x3-x2;
m0={{1,0,0,0},{0,1,0,0},{1,h0,h0^2,h0^3},{0,1,2*h0,3*h0^2}};
m1={{1,0,0,0},{0,1,0,0},{1,h1,h1^2,h1^3},{0,1,2*h1,3*h1^2}};
m2={{1,0,0,0},{0,1,0,0},{1,h2,h2^2,h2^3},{0,1,2*h2,3*h2^2}};
HermiteMat=ArrayFlatten[{{m0,0,0},{0,m1,0},{0,0,m2}}]

func=LinearSolve[HermiteMat]

f[x_]=x*Log[x];
fp[x_]=D[f[x],{x,1}]
Ans=func[{f[x0],fp[x0],f[x1],fp[x1],f[x1],fp[x1],f[x2],fp[x2],f[x2],fp[x2],f[x3],fp[x3]}];

Ans[[1]]

S[x_/;x0≤x≤x1] = Ans[[1]]+Ans[[2]]*(x-x0)+Ans[[3]]*(x-x0)^2+Ans[[4]]*(x-x0)^3;   
S[x_/;x1≤x≤x2] = Ans[[5]]+Ans[[6]]*(x-x1)+Ans[[7]]*(x-x1)^2+Ans[[8]]*(x-x1)^3;
S[x_/;x2≤x≤x3] = Ans[[9]]+Ans[[10]]*(x-x2)+Ans[[11]]*(x-x2)^2+Ans[[12]]*(x-x2)^3;
Plot[{f[x],S[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f[x]-S[x]},{x,x0,x3},PlotStyle->{Red}]

Sp[x_/;x0≤x≤x1] = Ans[[2]]+2*Ans[[3]]*(x-x0)+3*Ans[[4]]*(x-x0)^2;   
Sp[x_/;x1≤x≤x2] = Ans[[6]]+2*Ans[[7]]*(x-x1)+3*Ans[[8]]*(x-x1)^2;
Sp[x_/;x2≤x≤x3] = Ans[[10]]+2*Ans[[11]]*(x-x2)+3*Ans[[12]]*(x-x2)^2;
Plot[{fp[x],Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{fp[x]-Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]

Spp[x_/;x0≤x≤x1] = 2*Ans[[3]]+6*Ans[[4]]*(x-x0);   
Spp[x_/;x1≤x≤x2] = 2*Ans[[7]]+6*Ans[[8]]*(x-x1);
Spp[x_/;x2≤x≤x3] = 2*Ans[[11]]+6*Ans[[12]]*(x-x2);
Plot[{f''[x],Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f''[x]-Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]

ClampedSplineMat={
{0,1,0,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0,0,0,0},
{1,h0,h0^2,h0^3,0,0,0,0,0,0,0,0},
{0,1,2*h0,3*h0^2,0,-1,0,0,0,0,0,0},
{0,0,2,6*h0,0,0,-2,0,0,0,0,0},
{0,0,0,0,1,0,0,0,0,0,0,0},
{0,0,0,0,1,h1,h1^2,h1^3,0,0,0,0},
{0,0,0,0,0,1,2*h1,3*h1^2,0,-1,0,0},
{0,0,0,0,0,0,2,6*h1,0,0,-2,0},
{0,0,0,0,0,0,0,0,1,0,0,0},
{0,0,0,0,0,0,0,0,1,h2,h2^2,h2^3},
{0,0,0,0,0,0,0,0,0,1,2*h2,3*h2^2}
};

ClampedSolve=LinearSolve[ClampedSplineMat];
Ans=ClampedSolve[{fp[x0],f[x0],f[x1],0,0,f[x1],f[x2],0,0,f[x2],f[x3],fp[x3]}];

S[x_/;x0≤x≤x1] = Ans[[1]]+Ans[[2]]*(x-x0)+Ans[[3]]*(x-x0)^2+Ans[[4]]*(x-x0)^3;   
S[x_/;x1≤x≤x2] = Ans[[5]]+Ans[[6]]*(x-x1)+Ans[[7]]*(x-x1)^2+Ans[[8]]*(x-x1)^3;
S[x_/;x2≤x≤x3] = Ans[[9]]+Ans[[10]]*(x-x2)+Ans[[11]]*(x-x2)^2+Ans[[12]]*(x-x2)^3;
Plot[{f[x],S[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f[x]-S[x]},{x,x0,x3},PlotStyle->{Red}]

Sp[x_/;x0≤x≤x1] = Ans[[2]]+2*Ans[[3]]*(x-x0)+3*Ans[[4]]*(x-x0)^2;   
Sp[x_/;x1≤x≤x2] = Ans[[6]]+2*Ans[[7]]*(x-x1)+3*Ans[[8]]*(x-x1)^2;
Sp[x_/;x2≤x≤x3] = Ans[[10]]+2*Ans[[11]]*(x-x2)+3*Ans[[12]]*(x-x2)^2;
Plot[{fp[x],Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{fp[x]-Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]

Spp[x_/;x0≤x≤x1] = 2*Ans[[3]]+6*Ans[[4]]*(x-x0);   
Spp[x_/;x1≤x≤x2] = 2*Ans[[7]]+6*Ans[[8]]*(x-x1);
Spp[x_/;x2≤x≤x3] = 2*Ans[[11]]+6*Ans[[12]]*(x-x2);
Plot[{f''[x],Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f''[x]-Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]

NaturalSplineMat={
{0,0,2,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0,0,0,0},
{1,h0,h0^2,h0^3,0,0,0,0,0,0,0,0},
{0,1,2*h0,3*h0^2,0,-1,0,0,0,0,0,0},
{0,0,2,6*h0,0,0,-2,0,0,0,0,0},
{0,0,0,0,1,0,0,0,0,0,0,0},
{0,0,0,0,1,h1,h1^2,h1^3,0,0,0,0},
{0,0,0,0,0,1,2*h1,3*h1^2,0,-1,0,0},
{0,0,0,0,0,0,2,6*h1,0,0,-2,0},
{0,0,0,0,0,0,0,0,1,0,0,0},
{0,0,0,0,0,0,0,0,1,h2,h2^2,h2^3},
{0,0,0,0,0,0,0,0,0,0,2,6*h2}
};

NaturalSolve=LinearSolve[NaturalSplineMat];
Ans=NaturalSolve[{0,f[x0],f[x1],0,0,f[x1],f[x2],0,0,f[x2],f[x3],0}];

S[x_/;x0≤x≤x1] = Ans[[1]]+Ans[[2]]*(x-x0)+Ans[[3]]*(x-x0)^2+Ans[[4]]*(x-x0)^3;   
S[x_/;x1≤x≤x2] = Ans[[5]]+Ans[[6]]*(x-x1)+Ans[[7]]*(x-x1)^2+Ans[[8]]*(x-x1)^3;
S[x_/;x2≤x≤x3] = Ans[[9]]+Ans[[10]]*(x-x2)+Ans[[11]]*(x-x2)^2+Ans[[12]]*(x-x2)^3;
Plot[{f[x],S[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f[x]-S[x]},{x,x0,x3},PlotStyle->{Red}]

Sp[x_/;x0≤x≤x1] = Ans[[2]]+2*Ans[[3]]*(x-x0)+3*Ans[[4]]*(x-x0)^2;   
Sp[x_/;x1≤x≤x2] = Ans[[6]]+2*Ans[[7]]*(x-x1)+3*Ans[[8]]*(x-x1)^2;
Sp[x_/;x2≤x≤x3] = Ans[[10]]+2*Ans[[11]]*(x-x2)+3*Ans[[12]]*(x-x2)^2;
Plot[{fp[x],Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{fp[x]-Sp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]

Spp[x_/;x0≤x≤x1] = 2*Ans[[3]]+6*Ans[[4]]*(x-x0);   
Spp[x_/;x1≤x≤x2] = 2*Ans[[7]]+6*Ans[[8]]*(x-x1);
Spp[x_/;x2≤x≤x3] = 2*Ans[[11]]+6*Ans[[12]]*(x-x2);
Plot[{f''[x],Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]
Plot[{f''[x]-Spp[x]},{x,x0,x3},PlotStyle->{Red,Blue}]