Conditions for $df(X)$ to be a smooth field Given a vector field $X$ over a smooth manifold, under what conditions over $f$ is $df(X)$ a smooth field?
 A: The condition you're looking for is called $f$-relatedness:


*

*A smooth vector field $X$ on $M$ is $f$-related to a smooth vector field $Y$ on $N$ iff $df(X_p) = Y_{f(p)}$ for every $p \in M$.


If we regard $X\colon C^\infty(M) \to C^\infty(M)$ and $Y\colon C^\infty(N) \to C^\infty(N)$ as derivations, then it is a fact that $X$ and $Y$ are $f$-related iff for every smooth function $g\colon U \to \mathbb{R}$ (where $U\subset N$ is open), we have $X(g\circ f) = (Yg)\circ f$. ("Introduction to Smooth Manifolds" by John Lee, Lemma 4.8)
It is also a fact that if $f\colon M \to N$ is a diffeomorphism, then $df(X)$ will in fact be a smooth vector field on $N$. (As above, Proposition 4.10).
A: If $f$ is assumed to be an injective immersion, then $df(X)$ is a smooth vector field on $f(M)$ (it isn't necessarily true that it can be extended to all of $N$ though). This ties in with Jesse's answer because $df(X)$ is in fact the unique smooth vector field on $f(M)$ that is $f$-related to $X$ (see Proposition 8.27 in Lee's book).
