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 $F(987655)-F(987654)$.