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Re:[apcalculus] "Rule" for Logarithmic Differentiation??
Posted:
Oct 23, 2012 6:11 PM


NOTE: This apcalculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus Teacher Community Forum at https://apcommunity.collegeboard.org/gettingstarted and 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 multivariable > calculus techniques, toonot that I am recommending it > for 1st year calc.
For those interested in what Doug is alluding to, I've pasted a 2009 post of mine that goes into the details.
 
APCalculus post from 7 December 2009 http://mathforum.org/kb/message.jspa?messageID=6923571
This observation was recently made by John M. Johnson in his paper "Derivatives of generalized power functions" [Mathematics Teacher 102 #7 (March 2009), pp. 554557], and I've seen it in print in some other places as well:
Richard Katz and Stewart Venit, "Partial differentiation of functions of a single variable", Pi Mu Epsilon Journal 7 #6 (Spring 1982), 405406.
Gerry Myerson, "FFF #47: A natural way to differentiate an exponential", College Mathematics Journal 22 #5 (November 1991), p. 460.
G. E. Bilodeau, "An exponential rule", College Mathematics Journal 24 #4 (September 1993), 350351.
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 Journal 35 #5 (November 2004), 388390.
The expanded form of (d/dx)(U^V) can be explained by the multivariable chain rule. Let y = f(U,V), where U and V are differentiable functions of x. In this setting the chain rule takes the form
dy/dx = (del f)/(del U) * (del U)/(del x)
+ (del f)/(del V) * (del V)/(del x)
which equals
[V * U^(V1)] * (dU/dx) + [U^V * ln(U)] * (dV/dx)
when f(U,V) = U^V.
This exponential derivative identity was first published in 1695 by Leibniz, who also stated at this time that both he and Johann Bernoulli independently discovered it. See the following paper (freely available on the internet) for more historical issues relating to the derivative of a function to a function power.
Bos, Henk J. M. "Johann Bernoulli on Exponential Curves ...", Nieuw Archief voor Wiskunde (4) 14 (1996), 119. http://bernoulli.math.rug.nl/vorigelezingen/beginstuk/bos.pdf
You can also use the chain rule above to "explain" both the product rule and the quotient rule.
For instance, if y = f(U,V) = UV, then (del f)/(del U) = V and (del f)/(del V) = U, so
dy/dx = V * (dU/dx) + U * (dV/dx).
Also, if y = f(U,V) = U/V, then (del f)/(del U) = 1/V and (del f)/(del V) = U/(V^2), so
dy/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 to http://lyris.collegeboard.com/read/?forum=apcalculus



