Number of Partial Derivatives of Order n/Examples/Order 1 of 2 Variables
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Examples of Use of Number of Partial Derivatives of Order $n$
Let $u = \map f {x, y}$ be a real function of $2$ variables.
There are $2$ partial derivatives of $u$ of order $1$.
These are:
\(\ds \dfrac {\partial u} {\partial x}\) | \(=\) | \(\ds \map {f_1} {x, y}\) | ||||||||||||
\(\ds \dfrac {\partial u} {\partial y}\) | \(=\) | \(\ds \map {f_2} {x, y}\) |
Proof
From Number of Partial Derivatives of Order $n$, there are $m^n$ partial derivatives of order $n$ of a function of $m$ independent variables.
In this case, $m = 2$ and $n = 1$.
Thus there are $2^1 = 2$ partial derivatives of $u$ of order $1$.
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.3$ Higher Order Derivatives