Why Do Partitions Occur in Faà di Bruno ’ s Chain Rule For Higher Derivatives ?

It is well known that the coefficients in Faà di Bruno’s chain rule for n-th derivatives of functions of one variable can be expressed via counting of partitions. It turns out that this has a natural form as a formula for the vector case. Viewed as a purely algebraic fact, it is briefly “explained” in the first part of this note why a proof for this formula leads to partitions. In the rest of the note a proof for this formula is presented “from first principles” for the case of n-th Fréchet derivatives of mappings between Banach spaces. Again the proof “explains” why the formula has its form involving partitions.