Partial Derivatives

Partial derivatives measure the rate of change of a multivariable function with respect to one variable while holding others fixed. They generalize the derivative to functions $f : \mathbb{R}^n \to \mathbb{R}$ and are fundamental for optimization, differential equations, and physics. However, existence of partial derivatives does not guarantee differentiability — the total derivative (Fréchet derivative) is the correct generalization.

Definition

Definition9.1Partial derivative

Let $f : U \subseteq \mathbb{R}^n \to \mathbb{R}$ and $\mathbf{a} \in U$ . The partial derivative of $f$ with respect to $x_i$ at $\mathbf{a}$ is

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

where $\mathbf{e}_i$ is the $i$ -th standard basis vector.

RemarkOne variable at a time

Partial derivatives treat all variables except one as constants. They give the slope in the direction of the coordinate axes.

ExamplePartial derivatives of f(x, y) = x²y + y³

For $f(x, y) = x^2 y + y^3$ :

$\frac{\partial f}{\partial x} = 2xy, \quad \frac{\partial f}{\partial y} = x^2 + 3y^2.$

ExamplePartial derivatives exist but f not differentiable

Let $f(x, y) = \frac{xy}{x^2 + y^2}$ for $(x, y) \neq (0, 0)$ and $f(0, 0) = 0$ . Then:

$\frac{\partial f}{\partial x}(0, 0) = \lim_{h \to 0} \frac{f(h, 0)}{h} = 0$ .
$\frac{\partial f}{\partial y}(0, 0) = 0$ similarly.

However, $f$ is not continuous at $(0, 0)$ (approach along $y = x$ gives limit $1/2 \neq 0$ ), hence not differentiable. Existence of partial derivatives does not imply differentiability.

Gradient

Definition9.2Gradient

The gradient of $f : \mathbb{R}^n \to \mathbb{R}$ at $\mathbf{a}$ is the vector

$\nabla f(\mathbf{a}) = \left(\frac{\partial f}{\partial x_1}(\mathbf{a}), \ldots, \frac{\partial f}{\partial x_n}(\mathbf{a})\right).$

RemarkDirectional derivative

The gradient points in the direction of steepest ascent. The directional derivative in direction $\mathbf{v}$ is $D_{\mathbf{v}} f = \nabla f \cdot \mathbf{v}$ (when $f$ is differentiable).

ExampleGradient of f(x, y, z) = x²+ y² + z²

$\nabla f = (2x, 2y, 2z)$ . At $(1, 1, 1)$ , $\nabla f(1, 1, 1) = (2, 2, 2)$ points radially outward.

Higher-order partial derivatives

Definition9.3Second partial derivatives

$\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial}{\partial x_i}\left(\frac{\partial f}{\partial x_j}\right).$

Theorem9.1Clairaut's theorem (mixed partials)

If $\frac{\partial^2 f}{\partial x_i \partial x_j}$ and $\frac{\partial^2 f}{\partial x_j \partial x_i}$ exist and are continuous in a neighborhood of $\mathbf{a}$ , then

$\frac{\partial^2 f}{\partial x_i \partial x_j}(\mathbf{a}) = \frac{\partial^2 f}{\partial x_j \partial x_i}(\mathbf{a}).$

ExampleMixed partials for f(x, y) = x³y²

$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}(2x^3 y) = 6x^2 y = \frac{\partial}{\partial y}(3x^2 y^2) = \frac{\partial^2 f}{\partial y \partial x}$ .

Summary

Partial derivatives extend differentiation to multivariable functions:

Measure rate of change along coordinate directions.
Gradient $\nabla f$ points in direction of steepest ascent.
Mixed partials commute if continuous (Clairaut's theorem).
Existence of partials $\not\Rightarrow$ differentiability (need total derivative).

See Total Derivative for the correct notion of differentiability.