11.38 min read

Partial Derivative Functions

Just as the ordinary derivative $f'(x)$ is itself a function of $x$ , each partial derivative $\frac{\partial f}{\partial x_i}(\mathbf{x})$ is a function of the full vector $\mathbf{x}$ . We call these the partial derivative functions.

For $f: \mathbb{R}^n \to \mathbb{R}$ , we obtain $n$ partial derivative functions: $\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}$ , each mapping $\mathbb{R}^n \to \mathbb{R}$ .

We can then ask: are these partial derivative functions themselves continuous? Differentiable? Can we differentiate them again? This leads to higher-order partial derivatives such as $\frac{\partial^2 f}{\partial x_i \partial x_j}$ , which measures how the rate of change in direction $j$ itself varies in direction $i$ .

Formal View

Definition 11.4 — Partial Derivative Function

The $i$-th partial derivative function of

f: D \subseteq \mathbb{R}^n \to \mathbb{R}

is the function

\frac{\partial f}{\partial x_i}: D'\to \mathbb{R}

defined by

\frac{\partial f}{\partial x_i}(\mathbf{x}) = \lim_{h\to 0}\frac{f(\mathbf{x}+h\mathbf{e}_i)-f(\mathbf{x})}{h}

, where

D' \subseteq D

is the set of points where this limit exists.

Definition 11.5 — Mixed Partial Derivatives

The mixed second-order partial derivative

\frac{\partial^2 f}{\partial x_j \partial x_i}

is defined as

\frac{\partial}{\partial x_j}\left(\frac{\partial f}{\partial x_i}\right)

, i.e., first differentiate with respect to

x_i

, then with respect to

x_j

Clairaut's theorem: if $f$ is $C^2$ (all second partials are continuous), then mixed partials commute: $\partial^2 f/\partial x_j \partial x_i = \partial^2 f/\partial x_i \partial x_j$ .

Why This Matters

The regularity of partial derivative functions determines which calculus theorems apply and which numerical methods converge.

The Hessian matrix of second-order partial derivatives determines local curvature for optimization
Mixed partials appearing in physical laws (e.g., Maxwell equations) are assumed equal via Clairaut's theorem
Higher-order Taylor expansions in multiple variables require higher-order partial derivatives

Learning Resources

Higher Order Partial Derivatives

Khan Academy

Computing and interpreting second-order partial derivatives.

10 min

Clairaut's Theorem on Equality of Mixed Partials

Khan Academy

Why and when the order of partial differentiation can be switched.

8 min

Quiz

Question 1

For $f(x,y) = e^{x^2 y}$ , the mixed partial $\frac{\partial^2 f}{\partial y \partial x}$ equals:

Question 2

Clairaut's theorem states that mixed partial derivatives are always equal, regardless of any continuity conditions.

Common Mistakes

Applying Clairaut's theorem without checking continuity of the mixed partials.
Confusing the order of differentiation in $\partial^2 f / \partial x_j \partial x_i$ — the rightmost variable is differentiated first.
Forgetting that partial derivative functions are themselves functions of all variables, not just the differentiation variable.