{NotebookFileName[],DateString[]}

style[
t_
]:=
Style[t,Bold,FontSize->14,FontFamily->"Arial"]

cmarker[
size_:10,
pos_:{0,0}
]:=Graphics[Circle[pos],ImageSize->size];
cmarker[]

dmarker[
size_:10,
pos_:{0,0}
]:=Graphics[Disk[pos],ImageSize->size];
dmarker[]

$Assumptions={x∈Reals}

Mobius lesson

f1[
x_
]:=
x-(Sqrt[x])

Solve[0==f1'[x],x]

f2[
x_
]:=
((x^3)/(2-x))

Plot[{f2[x],f2'[x]},{x,-1,4}
,PlotLegends->"Expressions"
]

Curves with all derivatives 0

d0[
x_
]:=
(E^(-1/(x^2)))

d0[
x_/;x==0
]:=0

d0[0]

Plot[d0[x],{x,-1,1}
,Epilog->{
Inset[Graphics[cmarker[]],{0,0},{0,0},Background->White]
}
]

Plot[d0[x],{x,-10,10}
,PlotRange->{0,1}
]

Table[
D[d0[x],{x,n}],
{n,0,5}
]//Simplify

Does the derivative at 0 exist? At most, it has a removable discontinuity.

Table[
Limit[((D[d0[x],{x,n}])/x),x->0],
{n,0,20}
]

Solve[0==D[d0[x],{x,2}],x]

d0[x]/.Solve[0==D[d0[x],{x,2}],x]

Plot[x d0[x],{x,-1,1}
]

Plot[{x,x d0[x]},{x,-5,5}
,PlotLegends->"Expressions"
]

Table[
D[x d0[x],{x,n}],
{n,0,5}
]//Simplify

MVT

D[ArcTan[x],x]

FindRoot[ArcTan[2]-ArcTan[1]==(1/((c^2)+1)), {c,1.5}]

Solve[ArcTan[2]-ArcTan[1]==(1/((c^2)+1)),c]

c2=Select[c/.Solve[ArcTan[2]-ArcTan[1]==(1/((c^2)+1)),c],1<#<2&][[1]]

N[c2]

Block[{a=1,b=2,c=c2,f=ArcTan,expand=1.2,xmin,xmax,tanline},
xmin=((a+b)/2)-expand((b-a)/2);
xmax=((a+b)/2)+expand((b-a)/2);
tanline=(x|->f'[c](x-c)+f[c]);
Plot[{
f[x],
tanline[x]
},{x,xmin,xmax}
,PlotStyle->{Automatic,Red}
,PlotLegends->{"arctan(x)","tangent line at c≈1.45189"}
,Epilog->{
PointSize[Large],
Point[{{a,f[a]},{b,f[b]}}],
Line[{{a,f[a]},{b,f[b]}}],
Red,
Point[{{c,f[c]}}]
}
]
]