In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$
$$D_A F(B) = \frac{\mathrm{d}}{\mathrm{d}t}\bigg\rvert_{t=0} F(A+tB)\ .$$
I believe that it works, but I'm a little confused as to why: on the LHS of the equation, $B$ is a vector field, which $DF$ is allowed to act on since it is in its domain; on the RHS however, we feed $B$ to $F$, and it's not clear to me that $F$ knows what to do with it.
What am I not seeing? Is there some hidden identification of the manifold with its tangent space going on here? (In the case of the determinant map, $M \cong T_AM \cong \mathbb{R}^{n^2}$?) Is this formula always valid?
This formula only makes sense if $A+tB$ is still in $M$, so it's only going to work (in general) when $M$ is an affine space. This works, for example, if $M$ is an open subset of a vector space (which is the case with matrices or invertible matrices).