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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
Re:[ap-calculus] "Rule" for Logarithmic Differentiation??
Posted: Oct 23, 2012 6:11 PM
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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've
pasted a 2009 post of mine that goes into the details.


AP-Calculus post from 7 December 2009

This observation was recently made by John M. Johnson
in 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 differentiation
of functions of a single variable", Pi Mu Epsilon Journal
7 #6 (Spring 1982), 405-406.

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), 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 Journal
35 #5 (November 2004), 388-390.

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^(V-1)] * (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), 1-19.

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
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