Derivadas Utilizando Transformacion de Fourier

ϕ[x_] := Sin[x]

Plot[ ϕ[x], {x,-5,5}, PlotRange->All]

Fϕ[k_]:= FourierTransform[ϕ[x], x, k]

Fϕ[k]

df1x[x_] = InverseFourierTransform[I k  Fϕ[k], k,x]

Plot[ df1x[x], {x,-5,5}, PlotRange->All]

Funcion No-Derivable

ϕ[x_] := Exp[- Abs[x] ]

Plot[ ϕ[x], {x,-5,5}, PlotRange-> {0,1}]

dϕ[x_] :=D[ ϕ[x],x]

dϕ[x]

dϕ[1.0]

Fϕ[k_]:= FourierTransform[ϕ[x], x, k]

Fϕ[k]

Plot[((Sqrt[(2/π)])/(1+(k^2))) , {k,-5,5}]

Plot[ k ((Sqrt[(2/π)])/(1+(k^2))) , {k,-5,5}]

df1x[x_] = InverseFourierTransform[I k ((Sqrt[(2/π)])/(1+(k^2))), k,x]

Plot[ df1x[x], {x,-5,5}]