Partial Derivatives of Functions of Three or More Variables

Core Definition

For a scalar-valued function of three variables $w = f(x, y, z)$ defined on a domain $D \subset \mathbb{R}^3$ , the first-order partial derivatives are defined by holding two variables constant while taking the standard single-variable derivative with respect to the third:

\frac{\partial f}{\partial x}(x, y, z) = f_x = \lim_{h \to 0} \frac{f(x+h, y, z) - f(x, y, z)}{h}

\frac{\partial f}{\partial y}(x, y, z) = f_y = \lim_{h \to 0} \frac{f(x, y+h, z) - f(x, y, z)}{h}

\frac{\partial f}{\partial z}(x, y, z) = f_z = \lim_{h \to 0} \frac{f(x, y, z+h) - f(x, y, z)}{h}

More generally, for

f(x_1, x_2, \dots, x_n)

\frac{\partial f}{\partial x_i}

is computed by treating all

x_j

(

j \ne i

) as constants.

Concept Explanation

The Operational Rule in Higher Dimensions

When extending from two variables to three or more:

To find $\frac{\partial f}{\partial x}$ : Treat both $y$ and $z$ as constant numbers.
To find $\frac{\partial f}{\partial y}$ : Treat both $x$ and $z$ as constant numbers.
To find $\frac{\partial f}{\partial z}$ : Treat both $x$ and $y$ as constant numbers.

3D Spatial Domain (x, y, z) in R^3
   Step along x (parallel to x-axis, y and z held fixed) ===> Rate of change = f_x
   Step along y (parallel to y-axis, x and z held fixed) ===> Rate of change = f_y
   Step along z (parallel to z-axis, x and y held fixed) ===> Rate of change = f_z

Physical Interpretation in 3D Scalar Fields

Unlike functions of two variables whose graphs form 2D surfaces in 3D space, the graph of $w = f(x, y, z)$ exists in 4D space $\mathbb{R}^4$ , making direct graphical visualization impossible. Instead, engineers visualize $f(x, y, z)$ as a scalar field distributed throughout 3D space:

$T(x, y, z)$ : Temperature distribution inside an engine combustion chamber.
$P(x, y, z)$ : Acoustic pressure field inside an auditorium.
$\phi(x, y, z)$ : Electrostatic potential between capacitor plates.

In this context, the partial derivatives represent the directional spatial rates of change:

$\frac{\partial T}{\partial x}$ : Temperature change per unit distance moved east ( $x$ -axis).
$\frac{\partial T}{\partial y}$ : Temperature change per unit distance moved north ( $y$ -axis).
$\frac{\partial T}{\partial z}$ : Temperature change per unit distance moved upward ( $z$ -axis).

Together, these three components form the gradient vector field:

\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle

which points in the direction of maximum spatial increase and is orthogonal to the 3D Level Surfaces

f(x, y, z) = c

Engineering Applications

Combustion & Chemical Process Engineering: In rocket combustion chambers, spatial gradients of gas species concentrations $\frac{\partial C}{\partial x}, \frac{\partial C}{\partial y}, \frac{\partial C}{\partial z}$ govern molecular diffusion via Fick's laws.
Biomedical Imaging & Radiation Therapy: 3D radiation dose distributions $D(x, y, z)$ require evaluating partial derivatives across tumor boundaries to spare adjacent critical organs.
Electromagnetics & RF Engineering: Maxwell's equations evaluate spatial field derivatives $(\partial_x, \partial_y, \partial_z)$ of electric and magnetic fields to predict waveguide modes and antenna radiation patterns.

Example Problems

Example 1: Three-Variable Logarithmic Function with Radial Argument

Compute $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , and $\frac{\partial f}{\partial z}$ for:

f(x, y, z) = \ln(x^2 + y^2 + z^2)

Solution

Differentiate with respect to $x$ :
$\frac{\partial f}{\partial x} = \frac{1}{x^2+y^2+z^2} \cdot \frac{\partial}{\partial x}(x^2+y^2+z^2)$ Since $y$ and $z$ are constants, $\frac{\partial}{\partial x}(x^2+y^2+z^2) = 2x + 0 + 0 = 2x$ : $\frac{\partial f}{\partial x} = \frac{2x}{x^2+y^2+z^2}$
Differentiate with respect to $y$ and $z$ by symmetry:
$\frac{\partial f}{\partial y} = \frac{2y}{x^2+y^2+z^2}$ $\frac{\partial f}{\partial z} = \frac{2z}{x^2+y^2+z^2}$

Connections & References

2D Predecessor: Partial Derivatives
3D Level Surfaces: Level Surfaces
Vector Gradient: Session 12 - Gradient and Directional Derivatives
Course Sessions: Session 09 - Partial Derivatives