Let a point (i.e. the endpoint of $2$-dimensional Euclidean vector-valued differentiable function $f:[t_a\,..t_b]\rightarrow \mathbb R^2$ where $[t_a\,..t_b]\subseteq \mathbb R$) move on a circle. For simplicity assume that it doesn't stop or change direction of its movement. For any $t \in [t_a\,..t_b]$, let the angle between the line segment going from the circle's center to $f(t_a)$ and the line segment going from that center to $f(t)$ be written as $\theta(t)$.
Then we call the quantity $\dfrac{d\theta}{dt}$ at some $t_o$ the angular speed of that point at $t_0$.
How do we know that $\dfrac {d\theta}{dt}$ exists?
We know that $f$ is differentiable, but how do we know that $\theta$ is too?