Number of Distinct Partial Derivatives of Order n
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Theorem
Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \ldots, x_m$.
The number of distinct partial derivatives of $u$ of order $n$ is:
- $\dfrac {\paren {n + m - 1}!} {n! \paren {m - 1}!}$
That is, the same as the number of terms in a homogeneous polynomial in $m$ variables of degree $n$.
Proof
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Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.3$ Higher Order Derivatives