FDQR ( Freshman Dream Quotient Rule)

[JasMath33]

Homework Statement

I have been working with researching and writing a paper on the Freshman Dream Quotient Rule. This rule states

, 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.

 

[stevendaryl]

You can use the laws of derivatives to write:

(\frac{f}{g})' = \frac{f'}{g} - \frac{f g'}{g^2}

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

or multiplying through by g^2 g':

f' g g' - f (g')^2 = f' g^2

We can separate the f and g be rewriting as;

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}

So try picking some simple function for g and seeing if you can solve for f.

 

[pasmith]

If f(x) = e^{kx}, g(x) = e^{lx} then <br /> \left(\frac{f}{g}\right)&#039; = (k - l)e^{(k-l)x}, \\<br /> \frac{f&#039;}{g&#039;} = \frac kl e^{(k-l)x}. These can be made equal by suitable choice of k and l.

 

[mfb]

\frac{(g&#039;)^2}{g g&#039; - g^2} = \frac{g&#039;}{g} + \frac{g&#039;}{g&#039; - g}

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.