I'm working on a homework problem which seems obvious, but I am having a hard time proving/completing. The problem can be stated as follows:
Let $f,g:$ $\mathbb R$ $\rightarrow$ $\mathbb R$ be $\mathcal C^1$ functions and suppose they are topologically conjugated by a diffeomorphism $h$. Let $x_0$ be a periodic point of period $n$ of $f$. Prove that $y_0=h(x_0)$ is a periodic point of period $n$ of $g$, and that $(f^n)'(x_0)=(g^n)'(h(x_0))$
The first part seems pretty obvious from the definition of topological conjugacy - it preserves all the dynamics, so the periodic point is preserved as well. Anything beyond this seems to be lost on me, and I can't figure out how to start the second part.