## Blessed Francis Faa di Bruno - March 27

breski1 Francis, the last of 12 children, was born in northern Italy into an aristocratic family. He lived at a particularly turbulent time in history, when anti-Catholic and anti-papal sentiments …More

breski1 Francis, the last of 12 children, was born in northern Italy into an aristocratic family. He lived at a particularly turbulent time in history, when anti-Catholic and anti-papal sentiments were especially strong.

After being trained as a military officer, Francis was spotted by King Victor Emmanuel II, who was impressed with the young man’s character and learning. Invited by the king to tutor his two young sons, Francis agreed and prepared himself with additional studies. But with the role of the Church in education being a sticking point for many, the king was forced to withdraw his offer to the openly Catholic Francis and, instead, find a tutor more suitable to the secular state.

Francis soon left army life behind and pursued doctoral studies in Paris in mathematics and astronomy; he also showed a special interest in religion and asceticism. Despite his commitment to the scholarly life, Francis put much of his energy into charitable activities. He founded the Society of St. …More

After being trained as a military officer, Francis was spotted by King Victor Emmanuel II, who was impressed with the young man’s character and learning. Invited by the king to tutor his two young sons, Francis agreed and prepared himself with additional studies. But with the role of the Church in education being a sticking point for many, the king was forced to withdraw his offer to the openly Catholic Francis and, instead, find a tutor more suitable to the secular state.

Francis soon left army life behind and pursued doctoral studies in Paris in mathematics and astronomy; he also showed a special interest in religion and asceticism. Despite his commitment to the scholarly life, Francis put much of his energy into charitable activities. He founded the Society of St. …More

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook, considered the first published …More

Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook, considered the first published reference on the subject.

Perhaps the most well-known form of Faà di Bruno's formula says that

d n d x n f ( g ( x ) ) = ∑ n ! m 1 ! 1 ! m 1 m 2 ! 2 ! m 2 ⋯ m n ! n ! m n ⋅ f ( m 1 + ⋯ + m n ) ( g ( x ) ) ⋅ ∏ j = 1 n ( g ( j ) ( x ) ) m j , {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}

where the sum is over all n-tuples of nonnegative integers (m1, ..., mn) satisfying the constraint

1 ⋅ m 1 + 2 ⋅ m 2 + 3 ⋅ m 3 + ⋯ + n ⋅ m n = n . {\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}

Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

d n d x n f ( g ( x ) ) = ∑ n ! m 1 ! m 2 ! ⋯ m n ! ⋅ f ( m 1 + ⋯ + m n ) ( g ( x ) ) ⋅ ∏ j = 1 n ( g ( j ) ( x ) j ! ) m j . {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}

Combining the terms with the same value of m1 + m2 + ... + mn = k and noticing that mj has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials Bn,k(x1,...,xn−k+1):

d n d x n f ( g ( x ) ) = ∑ k = 1 n f ( k ) ( g ( x ) ) ⋅ B n , k ( g ′ ( x ) , g ″ ( x ) , … , g ( n − k + 1 ) ( x ) ) . {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}

Perhaps the most well-known form of Faà di Bruno's formula says that

d n d x n f ( g ( x ) ) = ∑ n ! m 1 ! 1 ! m 1 m 2 ! 2 ! m 2 ⋯ m n ! n ! m n ⋅ f ( m 1 + ⋯ + m n ) ( g ( x ) ) ⋅ ∏ j = 1 n ( g ( j ) ( x ) ) m j , {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}

where the sum is over all n-tuples of nonnegative integers (m1, ..., mn) satisfying the constraint

1 ⋅ m 1 + 2 ⋅ m 2 + 3 ⋅ m 3 + ⋯ + n ⋅ m n = n . {\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}

Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

d n d x n f ( g ( x ) ) = ∑ n ! m 1 ! m 2 ! ⋯ m n ! ⋅ f ( m 1 + ⋯ + m n ) ( g ( x ) ) ⋅ ∏ j = 1 n ( g ( j ) ( x ) j ! ) m j . {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}

Combining the terms with the same value of m1 + m2 + ... + mn = k and noticing that mj has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials Bn,k(x1,...,xn−k+1):

d n d x n f ( g ( x ) ) = ∑ k = 1 n f ( k ) ( g ( x ) ) ⋅ B n , k ( g ′ ( x ) , g ″ ( x ) , … , g ( n − k + 1 ) ( x ) ) . {\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=1}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}