Quiet@Remove["Global`*"];
$HistoryLength=2;
SetDirectory[NotebookDirectory[]];
Normal[Series[f[x],{x,x0,3}]]/.x->x0+Δx
Normal[Series[f[x],{x,x0,3}]]/.x->x0-Δx
%%-%

f=Sin;
Δx=0.1;
der1[x0_]:=((f[x0+Δx]-f[x0])/(Δx));
der2[x0_]:=((f[x0+Δx]-f[x0-Δx])/(2*Δx));
Plot[{f'[x]-der1[x],f'[x]-der2[x]},{x,0,2Pi},PlotStyle->{{Thick,Red},{Thick,Blue}},AxesLabel->{"x","err1  err2"},PlotLabel->"Δx= "<>ToString[Δx]]

Quiet@Remove["Global`*"];
vyr[x_]:=a*(x^2)+b*x+c;
resabc=Solve[{vyr[x0]==f0,vyr[x0-Δx]==fminus,vyr[x0+Δx]==fplus},{a,b,c}][[1]];
integrand=vyr[x]/.resabc;
Integrate[integrand,{x,x0-Δx,x0+Δx}]

derivace[{t_,y_}]:=(t/(1+(t^2)))*(1.+Sin[t+y]);
tmax=100;
nKroku=1500;
Δt=((1.*tmax)/nKroku);
krok[bod:{t_,y_}]:={t+Δt,y+Δt*derivace[bod]};
pocBod={0,1};
reseni=NestList[krok,pocBod,nKroku];
resNase=ListPlot[reseni];
rceProNDSolve=derivace[{t,y[t]}]==y'[t];
NDSolve[{rceProNDSolve,y[pocBod[[1]]]==pocBod[[2]]},y,{t,0,tmax}][[1]]
Plot[y[t]/.%,{t,0,tmax}];