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F(n) = F(n-1) + F(n-2) 

Is this differentiable? How or why not?

I understand how to differentiate equations like x**3 + 2x**2 + 3x + 5. But this is a recurrence relation and I don't know how it is differentiated.

I'm trying to understand the rate of growth of this equation. Can you tell me what would be the rate of growth at n = 987654?

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    I have a feeling what you're really looking for is asymptotics.2012-12-28

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Equations can never be differentiable, but functions may or may not be.

Without any context, it looks like your equation is actually a recurrence equation, and $n$ is assumed to be an integer. Functions that are only defined for integer values are never differentiable.

On the other hand, your equation defines the so called Fibonacci numbers (or something similar, depending on the initial values), and a resonable interpretation of "rate of growth" would be $F(n+1)-F(n)$, which can be computed. If $F(0) = F(1) = 1$, then $F(n) = \frac{1}{\sqrt 5}\left( \frac{1+\sqrt5}{2}\right)^n - \frac{1}{\sqrt 5}\left( \frac{1-\sqrt5}{2}\right)^n.$

I'll leave it to you to compute $F(987655)-F(987654)$.

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    Of course extensions are typically not unique, although equally "of course" everyone chooses the expression without the "noise", because it is simpler. And "undefined" is not the same as "does not exist", depending on one's philosophical and practical outlooks. And "not defined" does not entail "not definable", etc. Since I am not a formalist, these things are distinguishable to me, but, I recognize, potentially meaningless distinctions to others.2012-12-29