In the following paramterisation of the exponential distribution

$${\displaystyle f(t;\lambda )={\begin{cases}\lambda e^{-\lambda t}&t\geq 0,\\0&t<0.\end{cases}}},$$

$\lambda$ is called the "rate" parameter. If $T \sim \text{Exp}(\lambda)$, I think I understand the intuition behind why it's called a (average) rate - because it's the average number of arrivals per unit time $\left( \lambda = \frac{1}{\mathbb E(T)}\right)$; On average, there is 1 arrival in $\mathbb E (T)$ amount of time.

However, in some places (for example, in [continuous time markov chains](https://en.wikipedia.org/wiki/Transition_rate_matrix)), this $\lambda$ is called the **instantaneous** rate of change.

How is $\lambda$ an **instantaneous** rate of change (what makes it instantaneous?)?