I am trying to formulate an equation to determine the number of customers signed up to a subscription as a function of time. It's been 10 years since I studied maths, so bear with me!
The assumptions are: Customers will join at a constant rate. Customers will leave at a rate directly proportional to the number of customers currently subscribed. i.e. there when there are 200 customers, customers will be leaving at a rate double that of when there were 100 customers.
How do I write a function to describe this? What is it called when the rate of change is dependent on a dependent variable?
That is called a first-order differential equation. Let $y(t)$ be the number of customers at time $t$. Your conditions are that the rate of change of $y$, which is the derivative $y'$, is given by $y' = r - qy$, where $r$ is the number of customers that join and $q$ is the ratio of your current customers who leave in a unit of time. The solution to this equation is $y(t) = \frac{r}{q} - Ae^{-qt}$ for some constant $A$, which can be determined by the amount of customers at time $0$.
I.e., your number of customers will converge towards $r/q$ as time goes by, however many customers you start with.