Deriving Fa\`a di Bruno's formula for the derivative of a composite   function via compositions of integers
  We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer compositions, for which a Fa\`{a} di Bruno-like formula is quite naturally established. 
Steffen Eger
math.CO