Consider two manifolds M and N and a smooth map $F:M\rightarrow N$. Let's put that map aside for a minute and consider something else first.
Scenario 1:
We know that if $\exists$ $f:M\rightarrow \mathbb{R}$ we can pick an atlas, and consider curves in $M$ and tangent vectors to those curves and then take directional derivatives of $f$ at any given point $p\in M$ etc.
Scenario 2:
If we had another map $g: N\rightarrow\mathbb{R}$ and we wanted to take derivatives of $g$ at points in $N$, we could just do a similar thing like for $M$ and $f$ in Scenario 1.
Scenario 3:
Now let's get back to $F:M\rightarrow N$. We know that we can let $dF_p:T_pM\rightarrow T_{F(p)}N$ be the map between tangent spaces and that will be represented by the Jacobian matrix etc and we can use this structure to compute derivatives of $g:N\rightarrow \mathbb{R}$.
My question is this: If we want to take derivatives of $g:N\rightarrow \mathbb{R}$, why are we going through the trouble of Scenario 3 when we can have Scenario 2?