Given $f:S^1\to S^1: z\mapsto z^n$. I'm trying to show that $Df_p$ is orientation preserving: Let's try to show it at $p= (1,0)$. Tangent space at p is one dimensional and is spanned by basis vector $\partial/\partial y$. I assume $S^1$ to be oriented anticlockwise.
I want to show that $Df_p (\partial/\partial y)$ is a positive multiple of $\partial/\partial y$.
Take any $C^\infty(p)$ function $g$. Then
$Df_p(\partial/\partial y) g=\partial/\partial y|_p(g\circ f)=\partial/\partial\theta|_{\phi(p)}(g\circ f\circ \phi^{-1})$
How to go from here?