```Date: Oct 23, 2012 6:11 PM
Author: Dave L. Renfro
Subject: Re:[ap-calculus] "Rule" for Logarithmic Differentiation??

NOTE: This ap-calculus EDG will be closing in the next few weeks. Please sign up for the new AP CalculusTeacher Community Forum at https://apcommunity.collegeboard.org/getting-startedand post messages there. ------------------------------------------------------------------------------------------------Douglas J Kuhlmann wrote:> Barbara:  I think you missed the u' in Kristen's student's> formula.  She posted the same formula that you derived.> > On another note, one can prove this using multi-variable> calculus techniques, too--not that I am recommending it> for 1st year calc.For those interested in what Doug is alluding to, I'vepasted a 2009 post of mine that goes into the details.------------------------------------------------------------------------------------------------------------------------AP-Calculus post from 7 December 2009http://mathforum.org/kb/message.jspa?messageID=6923571This observation was recently made by John M. Johnsonin his paper "Derivatives of generalized power functions"[Mathematics Teacher 102 #7 (March 2009), pp. 554-557],and I've seen it in print in some other places as well:Richard Katz and Stewart Venit, "Partial differentiationof functions of a single variable", Pi Mu Epsilon Journal7 #6 (Spring 1982), 405-406.Gerry Myerson, "FFF #47: A natural way to differentiatean exponential", College Mathematics Journal 22 #5(November 1991), p. 460.G. E. Bilodeau, "An exponential rule", College MathematicsJournal 24 #4 (September 1993), 350-351.Dane W. Wu, "Miscellany", Pi Mu Epsilon Journal 10 #10(Spring 1999), 833.Noah Samuel Brannen and Ben Ford, "Logarithmic differentiation:Two wrongs make a right", College Mathematics Journal35 #5 (November 2004), 388-390.The expanded form of (d/dx)(U^V) can be explained by themultivariable chain rule. Let y = f(U,V), where U and Vare differentiable functions of x. In this setting thechain rule takes the formdy/dx = (del f)/(del U) * (del U)/(del x)+ (del f)/(del V) * (del V)/(del x)which equals[V * U^(V-1)] * (dU/dx) + [U^V * ln(U)] * (dV/dx)when f(U,V) = U^V.This exponential derivative identity was first publishedin 1695 by Leibniz, who also stated at this time that bothhe and Johann Bernoulli independently discovered it. Seethe following paper (freely available on the internet)for more historical issues relating to the derivative ofa function to a function power.Bos, Henk J. M. "Johann Bernoulli on Exponential Curves ...",Nieuw Archief voor Wiskunde (4) 14 (1996), 1-19.http://bernoulli.math.rug.nl/vorigelezingen/beginstuk/bos.pdfYou can also use the chain rule above to "explain" both theproduct rule and the quotient rule.For instance, if y = f(U,V) = UV, then (del f)/(del U) = Vand (del f)/(del V) = U, sody/dx = V * (dU/dx) + U * (dV/dx).Also, if y = f(U,V) = U/V, then (del f)/(del U) = 1/Vand (del f)/(del V) = -U/(V^2), sody/dx = (1/V) * (dU/dx) + [-U/(V^2)] * (dV/dx).= [V*(dU/dx) - U*(dV/dx)] / V^2------------------------------------------------------------------------------------------------------------------------Dave L. Renfro---To search the list archives for previous posts go tohttp://lyris.collegeboard.com/read/?forum=ap-calculus
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