Quiet@Remove["Global`*"];
$HistoryLength=2;
SetDirectory[NotebookDirectory[]];

taylor1=(Series[f[x],{x,x0,2}])//Normal
taylor1/.x->x0+Δx//Simplify
taylor1/.x->x0-Δx//Simplify
%%-%//Simplify

Series[Sin[x],{x,0,5}]

fce=Sin;
der1[x_,Δx_]:=((fce[x+Δx]-fce[x])/(Δx));
der2[x_,Δx_]:=((fce[x+Δx]-fce[x-Δx])/(2*Δx));
Plot[{((der1[x,0.1]-fce'[x])/(fce'[x])),((der2[x,0.1]-fce'[x])/(fce'[x]))},{x,0.1,1.4},PlotStyle->{{Thick,Red},{Thick,Blue}},PlotRange->All]

((der1[x,0.1]-fce'[x])/(fce'[x]))

Plot[(10^5)(E^(-x)) (-(E^x)+10.` (-(E^x)+(E^(0.1`+x)))),{x,0,1},PlotStyle->{Thick,Red}]