Is it true that, for every diffeomorphism $f:S\to S$ between a surface and itself, and for every pair of regular curves $\gamma_1, \gamma_2:(a,b)\to S$ tangent to each other at a point $p$, the curves $f\circ \gamma_1$ and $f\circ \gamma_2$ are tangent to each other at the point $f(p)$?
I do this:
Since $\gamma_1$ and $\gamma_2$ are tangent at $p$, assuming that $\gamma_1(0)=\gamma_2(0)=p$, then their velocities are parallel at $0$, i.e., there exists $c\in \mathbb{R}$ such that $\gamma_1'(0)=c\gamma_2'(0)$. Therefore, to show that $f\circ \gamma_1$ and $f\circ \gamma_2$ are tangent at $f(p)$, it is sufficient to demonstrate that their velocities are parallel at $0$. We have $(f\circ \gamma_1)'=df_p[\gamma_1'(0)]=df_p[c\gamma_2'(0)]$. Using the linearity of $df_p$, we obtain $df_p[c\gamma_2'(0)]=cdf_p[\gamma_2'(0)]=c(f\circ \gamma_2)'$. Hence, we have concluded.
However, I haven't understood where the assumption that $f$ is a diffeomorphism is used, or better, how this fact should be employed. It tells us that $df_p$ is an isomorphism, but where should this fact be utilized?