Write a function that returns the logarithmic derivative

Here's another way to proceed, using Derivative[], and sidestepping the use of a dummy variable:

LogDerivative[f_] := Derivative[1][Composition[Log, f]]

Test:

LogDerivative[Sin][x]
   Cot[x]

LogDerivative[Gamma][x]
   PolyGamma[0, x]

LogDerivative[#^3 &][x]
   3/x

Your operator must depend on both function and variable - in analogy to D function:

logD[f_, x_] := D[f, x]/f

or an alternative definition:

logD[f_, x_] := D[Log[f], x]

Of course your variables of differentiation and in the function must agree. Test it:

logD[f[x], x]

Derivative[1][f][x]/f[x]

logD[Sin[x], x]

Cot[x]

f = x^2; logD[f, x]

2/x

g[y_] := Exp[1/y]; logD[g[y], y]

-(1/y^2)

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