If $M,N$ are two submanifolds of e.g. $\R^n$ and $F:M\rar N$ is smooth, i.e. $F$ is the restriction of a smooth map from an open superset of $M$ into $\R^n$ such that $F(M)\sbe N$, then it's derivative $DF(x)$ at any point $x\in M$ maps the tangent space $T_xM$ of $M$ at $x$ into the tangent space $T_{F(x)}N$.

There are several definitions of the tangent space $T_xM$ of a submanifold $M$ of $\R^n$ at the point $x$; for our purpose the handiest definition is the following: $T_xM$ is the space of all derivavtives $c^\prime(0)$ of smooth curves $c:(-\d,\d)\rar M$ such that $c(0)=x$. Then by the chain-rule $(F\circ c)^\prime(0)=DF(c(0))c^\prime(0)$, i.e. $DF(x)c^\prime(0)\in T_{F(x)}N$.
The catch is that this definition of the tangent space doesn't implicate easily that $T_xM$ is a vector-space.