Faà di Bruno's Formula/Lemma 1

Theorem

Let $m \in \Z_{\ge 1}$ be a (strictly) positive integer.

Let $k_m \in \Z_{\ge 1}$ also be a (strictly) positive integer.

Let $u: \R \to \R$ be a function of $x$ which is appropriately differentiable.

Then:

$\map {D_x} {\paren {D_x^m u}^{k_m} } = k_m \paren {D_x^m u}^{k_m - 1} D_x^{m + 1} u$

Proof

\(\ds \map {D_x} {\paren {D_x^m u}^{k_m} }\)

\(=\)

\(\ds \paren {k_m \paren {D_x^m u}^{k_m - 1} } \map {D_x} {D_x^m u}\)

Derivative of Power of Function

\(\ds \)

\(=\)

\(\ds k_m \paren {\paren {D_x^m u}^{k_m - 1} } D_x^{m + 1} u\)

Definition of Higher Derivative

$\blacksquare$