Suppose we have a recurrence relation of the form :
$$ a_{n+1} = \phi(a_n)$$
Where, $a_n \in R$,
I believe we can make statements about the recurrence, say eg, boundedness, monotonicity and so on, by analyzing $\phi$ as a single variable function on a suitable domain, and see it's derivatives. Am I onto something here?
Short answer, "yes"; longer answer, "yes, but you're not the first".
This type of sequence is called an "iterated function system". You're right that the derivative is important; it can be used to determine the stability of a fixed point of the system.
As a concrete example, you might want to solve an equation with this type of iterated system. Say you want to solve $$x^3-6x^2+11x-6=0$$ (which, obviously, you can solve other ways, but we can use it as an example)
You can rewrite this as $$x = \frac16 \left(x^2+11-\frac{6}{x}\right)$$
which you can think of as your type of sequence by writing $$\phi(x) = \frac16 \left(x^2+11-\frac{6}{x}\right)$$
and defining $x_{n+1} = \phi\left(x_n\right)$
Fixed points of $\phi$ (ie those $x$ such that $\phi(x)=x$) are precisely the roots of the cubic (ie $1,2,3$). If we iterate the system for different starting points, which of these roots does it converge to? How is this related to the value of the derivative of $\phi$ at those points? Does it always converge? Can we rewrite the original equation in different ways to converge to a different root?
Of course, we can use this idea to solve more complicated equations too.
Some things you might want to look up are: iterated function systems, fixed point stability, fixed point iteration, Newton's method. Fractals based on these systems include bifurcation diagrams, Julia sets, the Mandelbrot set Barnsley's fern.
Hope that's useful!