dery=F[y[t],t];
dos1={y'[t]->dery};
D[dery,t]/.dos1
D[%,t]/.dos1

dery=((y[t])/(1+((y[t])^2)))*Sin[t+3y[t]];
y0=1.;
tmax=10;
sol=NDSolve[{y'[t]==dery,y[0]==y0},y,{t,0,tmax}][[1]];

pl1=Plot[y[t]/.sol,{t,0,tmax}];
nKroku=1000;
Δt=(tmax/nKroku);
dery1[{t_,y_}]:=(y/(1+(y^2)))*Sin[t+3y];
krok1Rad[bod:{t_,y_}]:={t+Δt,y+dery1[bod]};
pocStav={0,y0};
res1=NestList[krok1Rad,pocStav,nKroku];
pl2=ListPlot[res1,PlotStyle->{Red,PointSize[0.01]}];
Show[pl1,pl2,PlotRange->All];

der2[{t_,y_}]=(D[dery1[{t,y[t]}],t]/.y'[t]->dery1[{t,y[t]}])/.y[t]->y;

krok1Rad[bod:{t_,y_}]:={t+Δt,y+dery1[bod]};
krok2Rad[bod:{t_,y_}]:={t+Δt,y+dery1[bod]+(1/2.)*der2[bod]};
res2=NestList[krok2Rad,pocStav,nKroku];
pl3=ListPlot[res2,PlotStyle->{Black,PointSize[0.01]}];
Show[pl1,pl2,pl3,PlotRange->All]