Let $\alpha(t)\in\{0,1\}: 0\leq t\leq 1$ be a two state continuous Markov chain with the generator (https://en.wikipedia.org/wiki/Continuous-time_Markov_chain) $$ Q=\begin{bmatrix} -\alpha & \alpha\\ \beta &-\beta\\\ \end{bmatrix} $$ Assume that $\alpha(t)=i \in\{0,1\}$. Let $$f(t)=\int_t^1\mathbb{I}_{\{\alpha(u)=1\}}(u) du $$ where $\mathbb{I}$ is the indicator function.
My questions:
1) How to find $\frac{df}{dt}$
2) Can we find the density function of $f(t)$ ?