Definition of tangent spaces independent of coordinate patches

Question

Let $A\subset V$ be a manifold and $f:B\rightarrow C\subset A$ a coordinate patch. Here $B\subset W$ is an open subset of some vector space $W$.

1. Let $f':B'\rightarrow C'$ be another coordinate patch and $a\in C'$ s.t. $f'(b')=a$. How do I prove that $F=f^{-1}\circ f'$ is a diffeomorphism and $DF(b'):W\rightarrow W$ is a linear isomorphism?
2. How do I show that $Df'(b')=Df(b)\circ DF(b')$?
3. Why does 2. imply im$Df(b')=$im$Df(b)$?

WHAT I KNOW:

I know that the tangent space $T_aA$ is the im$Df(b)$ where $f(b)=a\in A$.
Further, for 1., I will need to restrict the domains and ranges, such that $f^{-1}\circ f'$ can be well defined. The restriction gives us $f^{-1}\circ f'|_{C\cap C'}$.