The Baker-Campbell-Hausdorff (BCH) formula links the Lie group and Lie algebra by giving a formula for group multiplication of elements close to the identity in terms of the Lie bracket. This has applications in geometry, algebra, and partial differential equations, and provides simple proofs of further developments of Lie theory. To date, there is no BCH formula for the diffeomorphism groups, although they are infinite dimensional analogues of Lie groups, because the intrinsic group exponential map is not even locally surjective. In this paper we provide the relevant background in an expository manner, starting with the finite dimensional case, and then take a first step towards developing a BCH formula for the exponential map of a right-invariant metric on a Lie Group G by deriving an explicit formula for its derivative, in the spirit of Poincaré. We then identify features that allow for the finite dimensional results to lift to the infinite dimensional setting of the Diffeomorphism groups and reprove our finite dimensional results on the diffeomorphism groups. Along the way we discuss additional applications of our formulas and conclude with a program to complete the BCH formula.