it is clear (geometrically) that ${\mathbb S}^1$ is parallelizable because we can consider a vector field of unit tangent vectors at each point (say, in anti-clockwise direction).

I am having trouble understanding how to write down the above argument mathematically

....we need to come up with a smooth vector field $X: {\mathbb S}^1 -> T{\mathbb S}^1$, so that $X(p) = X^1(x) d/dx$.

what is the choice for a smooth function $X^1(x)$ to have the vector field as above??

can we consider the following (coordinate) vector field $X(p) = d/dx$??