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Topic: RE:[ap-calculus] Re:a paradox
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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
RE:[ap-calculus] Re:a paradox
Posted: Dec 7, 2009 11:20 AM
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Bradley Stoll wrote:

> Here's an interesting thing that took me 12 years of
> teaching ap calculus to "see" and I only discovered
> it because a student did this:
> I've always used logarithmic diff to find the deriv
> with respect to x of f^g, where f and g are both
> functions of x And, I've always stopped after multiplying
> both sides by f^g, not distributing. Thus, the answer
> ended at:

censor foil

> (f^g)' = (g'ln(f) + gf '/f)f^g. And, that is where I
> stopped. However, the intersting part appears only if
> you distribute, to find that:
> (f^g)' = g f^(g-1)f ' + f^g ln(f) g '
> So, taking the derivative straight from f^g can be thought
> of in this manner: treat f as a function and g as a constant;
> use the power rule, and of course the chain rule. Then,
> treat f as the constant and g as the function and take
> the derivative, again using the chain rule. Then, add the
> two together. I suppose that this sort of makes
> least up to the part where I say, "then add the two together".

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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