A new derivative

We can define a logarithmic derivative of a function \(f(x)\) as \[ f^\ddagger (x) = \frac{d}{dx} ln \; f(x) = \frac{f'(x)}{f(x)}\]

It's easy to see that it takes on a few nicer properties than derivatives typically do when it comes to quotients and composition

\[ (fg)^\ddagger = f^\ddagger + g^\ddagger \] \[ (f/g)^\ddagger = f^\ddagger - g^\ddagger \] \[ (f(g))^\ddagger = f^\ddagger (g) g' \]

We can define common derivatives in terms of the logarithmic derivative:

\[ (x^n)^\ddagger = n \frac{1}{x} \] \[ (e^x)^\ddagger = 1 \] \[ (cos(x))^\ddagger = -tan(x) \] \[ (tan(x))^\ddagger = tan(x) + cot(x) \]

Can you find a function \(f(x)\) such that \(f^\ddagger (x) = f(x)\)?

Note by Levi Adam Walker
2 weeks ago

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Yes.........there are infinitely many functions.......You need to specify the boundary conditions for a unique solution........Otherwise, it is a simple differential equation......

Aaghaz Mahajan - 1 week, 6 days ago

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It's just an exercise in finding a solution, not the solution :)

Levi Adam Walker - 1 week, 6 days ago

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Ohh.........yup.....didn't see that...:P

Aaghaz Mahajan - 1 week, 6 days ago

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Would be nice if you changed the notation for the logarithmic derivative... maybe \(f_{\text{L}}(x)\) or \(Lf(x)\). As Aaghaz has said somewhere else, a solution would come from a family of solutions to the differential equation \(y' = y^2\). Have a look here for more information on this stuff.

Gennady Notowidigdo - 1 week, 6 days ago

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I've seen \( \ddagger \) used, so I just adopted that notation. I also find it more aesthetically pleasing tbh, and we all know I'm a sucker for aesthetics.

Levi Adam Walker - 1 week, 6 days ago

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