FDQR ( Freshman Dream Quotient Rule)

  • #1
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Homework Statement


I have been working with researching and writing a paper on the Freshman Dream Quotient Rule. This rule states
upload_2016-6-27_9-14-3.png
, and I was wondering if anyone can come up with an example of 2 functions which make this work.

Homework Equations




The Attempt at a Solution


I have looked at proofs and everything of this, but have not been able to come up with an example for my paper. For all I know, there may not be an example.
 

Answers and Replies

  • #2
stevendaryl
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You can use the laws of derivatives to write:

[itex](\frac{f}{g})' = \frac{f'}{g} - \frac{f g'}{g^2}[/itex]

So you're trying to solve:
[itex]\frac{f'}{g} - \frac{f g'}{g^2} = \frac{f'}{g'}[/itex]

or multiplying through by [itex]g^2 g'[/itex]:

[itex]f' g g' - f (g')^2 = f' g^2[/itex]

We can separate the [itex]f[/itex] and [itex]g[/itex] be rewriting as;

[itex]f'(g g' - g^2) = f (g')^2 \Rightarrow \frac{f'}{f} = \frac{(g')^2}{g g' - g^2} = \frac{g'}{g} + \frac{g'}{g' - g}[/itex]

So try picking some simple function for [itex]g[/itex] and seeing if you can solve for [itex]f[/itex].
 
  • #3
pasmith
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If [itex]f(x) = e^{kx}[/itex], [itex]g(x) = e^{lx}[/itex] then [tex]
\left(\frac{f}{g}\right)' = (k - l)e^{(k-l)x}, \\
\frac{f'}{g'} = \frac kl e^{(k-l)x}.[/tex] These can be made equal by suitable choice of [itex]k[/itex] and [itex]l[/itex].
 
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  • #4
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[itex] \frac{(g')^2}{g g' - g^2} = \frac{g'}{g} + \frac{g'}{g' - g}[/itex]
That looks like a freshman's dream on its own but it is correct.
That approach leads to a nice solution for some simple polynomial g.
 

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