ClearAll

x0[t]= x[t]
x1[t]=f[x[t],t]
x2[t]=D[x1[t],t] /. x'[t] -> f[x[t],t]
x3[t]=D[x2[t],t] /. x'[t] -> f[x[t],t]
x4[t]=D[x3[t],t] /. x'[t] -> f[x[t],t]
x5[t]=D[x3[t],t] /. x'[t] -> f[x[t],t]
x6[t]=D[x3[t],t] /. x'[t] -> f[x[t],t]

xx2=x2[t] /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
xx3=x3[t] /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
xx4=x4[t] /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
xx5=x5[t] /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
xx6=x6[t] /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}

K1=f[x[t],t]
K2=f[x[t]+h*a21*K1,t+h*c2]
K3=f[x[t]+h*(a31*K1+a32*K2),t+h*c3]
K4=f[x[t]+h*(a41*K1+a42*K2+a43*K3),t+h*c4]
K5=f[x[t]+h*(a51*K1+a52*K2+a53*K3+a54*K4),t+h*c5]
K6=f[x[t]+h*(a61*K1+a62*K2+a63*K3+a64*K4+a65*K5),t+h*c6]
y[h]=x[t]+h*(b1*K1+b2*K2+b3*K3+b4*K4+b5*K5+b6*K6)
y1=Simplify[D[y[h],h] /. h->0]
y2=Simplify[D[y[h],{h,2}] /. h->0]
*y3=Simplify[D[y[h],{h,3}] /. h->0]
*y4=Simplify[D[y[h],{h,4}] /. h->0]
*y5=Simplify[D[y[h],{h,5}] /. h->0]
*y6=Simplify[D[y[h],{h,6}] /. h->0]

yy1=y1 /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
yy2=y2 /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
yy3=y3/. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}
yy4=y4 /. {f[x[t],t] -> F[0,0],Derivative[1,0][f][x[t],t] -> F[1,0], Derivative[0,1][f][x[t],t] -> F[0,1], Derivative[2,0][f][x[t],t] -> F[2,0], Derivative[1,1][f][x[t],t] -> F[1,1], Derivative[0,2][f][x[t],t] -> F[0,2], Derivative[3,0][f][x[t],t] -> F[3,0], Derivative[2,1][f][x[t],t] -> F[2,1], Derivative[1,2][f][x[t],t] -> F[1,2], Derivative[0,3][f][x[t],t] -> F[0,3]}

xx1=F[0,0]
z1=Simplify[xx1-yy1]
z2=Simplify[xx2-yy2]
z3=Simplify[xx3-yy3]
z4=Simplify[xx4-yy4]

Do[Do[Do[Do[Do[Do[ {Z=Simplify[(1/i!)*(1/j!)*D[D[z1,{F[m,n],i}],{F[m1,n1],j}] /. {F[0,0]->0, F[1,0]->0, F[0,1]->0, F[2,0]-> 0,  F[1,1]-> 0, F[0,2]-> 0, F[3,0]-> 0, F[2,1]-> 0, F[1,2]-> 0, F[0,3]-> 0} ],Print[Z]},{i,j,3}],{j,0,3}],{m,0,3}],{n,0,3}],{m1,0,3}],{n1,0,3}]
Do[Do[Do[Do[Do[Do[ {Z=Simplify[(1/i!)*(1/j!)*D[D[z2,{F[m,n],i}],{F[m1,n1],j}] /. {F[0,0]->0, F[1,0]->0, F[0,1]->0, F[2,0]-> 0,  F[1,1]-> 0, F[0,2]-> 0, F[3,0]-> 0, F[2,1]-> 0, F[1,2]-> 0, F[0,3]-> 0} ],Print[Z]},{i,j,3}],{j,0,3}],{m,0,3}],{n,0,3}],{m1,0,3}],{n1,0,3}]
Do[Do[Do[Do[Do[Do[ {Z=Simplify[(1/i!)*(1/j!)*D[D[z3,{F[m,n],i}],{F[m1,n1],j}] /. {F[0,0]->0, F[1,0]->0, F[0,1]->0, F[2,0]-> 0,  F[1,1]-> 0, F[0,2]-> 0, F[3,0]-> 0, F[2,1]-> 0, F[1,2]-> 0, F[0,3]-> 0} ],Print[Z]},{i,j,3}],{j,0,3}],{m,0,3}],{n,0,3}],{m1,0,3}],{n1,0,3}]
Do[Do[Do[Do[Do[Do[ {Z=Simplify[(1/i!)*(1/j!)*D[D[z4,{F[m,n],i}],{F[m1,n1],j}] /. {F[0,0]->0, F[1,0]->0, F[0,1]->0, F[2,0]-> 0,  F[1,1]-> 0, F[0,2]-> 0, F[3,0]-> 0, F[2,1]-> 0, F[1,2]-> 0, F[0,3]-> 0} ],Print[Z]},{i,j,3}],{j,0,3}],{m,0,3}],{n,0,3}],{m1,0,3}],{n1,0,3}]