D[t^2+t^3*Cos[t], t]

D[t^2+t^3*Cos[t], {t, 4}]

tangent[gamma_][t_]:=Simplify[
D[gamma[t],t]/(D[gamma[t],t].D[gamma[t],t])^(1/2)]

cub1[t_]:={3t-t^3, 3t^2, 3t+t^3}

tangent[cub1][t]

trifoglioprb[t_ ]:={Cos[3t]Cos[t],Cos[3t]Sin[t],Cos[3t]^2}

ParametricPlot3D[trifoglioprb[t], {t, 0, 4Pi}]

tangent[trifoglioprb][t]

ParametricPlot3D[{trifoglioprb[t],tangent[trifoglioprb][t]} //Evaluate,  {t, 0, 4Pi}]

spiralelog3D[a_, b_, c_][t_]:= {a*E^(b*t)*Cos[t], a*E^(b*t)*Sin[t], c*t }

ParametricPlot3D[spiralelog3D[1, 0.3, .4][t], {t, 0, 10Pi}, PlotRange -> 1000]

tangent[spiralelog3D[1, .08, .4]][t]

ParametricPlot3D[{spiralelog3D[1, .08, .4][t],tangent[spiralelog3D[1, .08, .4]][t]} //Evaluate,  {t, 0, 4Pi}]

binormal[gamma_][t_]:=Simplify[Cross[D[gamma[t],t],D[gamma[t],{t,2}]]/(Cross[D[gamma[t],t],D[gamma[t],{t,2}]].Cross[D[gamma[t],t],D[gamma[t],{t,2}]])^(1/2)]

normal[gamma_][t_]:=Simplify[Cross[binormal[gamma][t],tangent[gamma][t]]]

curvatura[gamma_][t_]:= Simplify[(Cross[D[gamma[t],t],D[gamma[t],{t,2}]].Cross[D[gamma[t],t],D[gamma[t],{t,2}]])^(1/2)/(D[gamma[t],t].D[gamma[t],t])^(3/2)]

torsione[gamma_][t_]:=Simplify[(Cross[D[gamma[t],t],D[gamma[t],{t,2}]].D[gamma[t],{t,3}])/(Cross[D[gamma[t],t],D[gamma[t],{t,2}]].Cross[D[gamma[t],t],D[gamma[t],{t,2}]])]

ParametricPlot3D[{Cos[t], Sin[t], Sin[2t]}, {t, 0, 2Pi}]

compito[t_] := {Cos[t], Sin[t], Sin[2t]}

tangent[compito][t]

curvatura[compito][t]

torsione[compito][t]

Det[{ D[compito[t],t], D[compito[t], {t,2}], D[compito[t], {t, 3}]}//MatrixForm]

Det[(({{-Sin[t], Cos[t], 2 Cos[2 t]}, {-Cos[t], -Sin[t], -4 Sin[2 t]}, {Sin[t], -Cos[t], -8 Cos[2 t]}}))]

Simplify[Out[42]]

Plot[{curvatura[trifoglioprb][t], torsione[trifoglioprb][t]} //Evaluate, {t, 0, 2Pi}, PlotStyle -> {Red, Black}]

Plot[{curvatura[compito][t], torsione[compito][t]} //Evaluate, {t, 0, 2Pi}, PlotStyle -> {Red, Black}]

spiralelog3D[1, .08, .4]

Plot[{curvatura[spiralelog3D[1, .08, .4]][t], torsione[spiralelog3D[1, .08, .4]][t]} //Evaluate, {t, 0, 100Pi}, PlotStyle -> {Red, Black}]