Who was the first person to use logarithmic differentiation? This is a math history question.  And I'm curious if it was Euler or someone else. In what mathematical work did it first appear?  I don't have the resources/resourcefulness to answer this question.
 A: I doubt it's the earliest, but the method does appear in Euler's works:


  
*Exposita logarithmorum differentiatone, progrediamur ad quantitates exponentiales, seu eiusmodi potestates, quarum exponentes sint variabiles.  Huiusmodi autem ipsius $x$ functionum differentialia per logarithmorum differentiationem inveniri possunt hoc modo.  Quaeratur differentiale ipsius $a^x$, ad quod investigandum ponator $y=a^x$, eritque logarithmis sumendis $ly = xla$.  Sumantur iam differentialia, atque obtinebitur $\frac{dy}{y} = dxla$; unde sit $dy = ydxla$, cum autem sit $y=a^x$, erit $dy = a^x dxla$, quod est differentiale ipsius $a^x$.  Simili modo, si sit $p$ functio quaecunque ipsius $x$, huius quantitatis exponentialis $a^p$ differentiale erit $= a^p dpla$.
  

This is from chapter 6 of volume 1 of his textbook Institutiones calculi differentialis (starting on page 8 of the pdf linked there), from around 1750.  It's in Latin, of course, but Euler's notation is a lot like ours (or rather, ours is a lot like his), so you can probably figure out what's he's presenting above.
(The next few sections give other applications of the method, and some alternative methods for the same problems.)
