Partial Derivatives

One-line statement: A partial derivative measures the instantaneous rate of change of a multivariable function with respect to one variable while all other variables are held fixed. Key result in words: For

f (x, y)

, the partial

\partial f / \partial x

is the ordinary derivative obtained by treating

y

as a constant. Where it applies: Multivariable calculus, vector calculus, optimization, thermodynamics, PDEs, economics, machine learning (gradients). Who/when: Notation and systematic theory developed in the 18th–19th centuries; the curly

\partial

symbol popularized by Legendre (1786) and Jacobi (1841).

A partial derivative generalizes the single-variable derivative to functions of several variables. For a function $f : R^{n} \to R$ , the partial derivative with respect to a variable $x_{i}$ records how $f$ changes as $x_{i}$ varies infinitesimally while the remaining variables are frozen. It matters because most quantities in science depend on many variables simultaneously — temperature varies in space and time, profit depends on price and quantity — and partial derivatives isolate the influence of each input. They are the building blocks of the gradient, the Jacobian, and the entire apparatus of multivariable analysis. A surprising fact: a function can have all partial derivatives at a point yet still fail to be continuous there — partial differentiability is weaker than the full notion of differentiability.

The Core Insight

Freeze everything but one variable, then differentiate normally. A partial derivative reduces a hard multivariable problem to a familiar single-variable one by slicing the domain along a coordinate axis. Geometrically, you intersect the surface

z = f (x, y)

with a plane parallel to a coordinate axis and measure the slope of the resulting curve. Every rule from single-variable calculus (product, quotient, chain) applies verbatim once the other variables are treated as constants.

Formal Statement

Partial Derivative

Let $f : U \subseteq R^{n} \to R$ be defined on an open set $U$ , and let $a = (a_{1}, \dots, a_{n}) \in U$ . The partial derivative of $f$ with respect to $x_{i}$ at $a$ is

\frac{\partial f}{\partial x _{i}} (a) = h \to 0 lim \frac{f ( a _{1} , \dots , a _{i} + h , \dots , a _{n} ) - f ( a _{1} , \dots , a _{i} , \dots , a _{n} )}{h}

⟳This is the ordinary difference quotient in the single direction

e_{i}

, holding all other coordinates fixed at their values

a_{j}

(

j \neq = i

Here every symbol is: $f$ is the scalar-valued function; $a$ is the base point; $x_{i}$ is the differentiation variable; $h$ is the increment; $e_{i}$ is the $i$ -th standard basis vector. Common notations include $\partial f / \partial x_{i}$ , $f_{x_{i}}$ , $\partial_{x_{i}} f$ , and $D_{i} f$ .

Equivalent Directional Form

The partial derivative is the directional derivative along a coordinate axis:

\frac{\partial f}{\partial x _{i}} (a) = h \to 0 lim \frac{f ( a + h e _{i} ) - f ( a )}{h}

⟳Replace the

i

-th coordinate increment in [1] with the vector form

a + h e_{i}

, which changes only the

i

-th component.

Clairaut–Schwarz Theorem (Equality of Mixed Partials)

If $f : U \to R$ has continuous second-order partial derivatives $\partial^{2} f / \partial x \partial y$ and $\partial^{2} f / \partial y \partial x$ in a neighborhood of $a$ , then they are equal:

\frac{\partial ^{2} f}{\partial x \partial y} (a) = \frac{\partial ^{2} f}{\partial y \partial x} (a)

⟳Apply the Mean Value Theorem twice to the second difference

Δ = f (a + h, b + k) - f (a + h, b) - f (a, b + k) + f (a, b)

, expressing it through both orders of differentiation, then take

h, k \to 0

using continuity.

Differentiability Implies Partials Exist

If $f$ is (totally) differentiable at $a$ — meaning there exists a linear map $L$ with $f (a + h) = f (a) + L h + o (∥ h ∥)$ — then all partial derivatives exist at $a$ , and the total differential is

df = i = 1 \sum n \frac{\partial f}{\partial x _{i}} d x_{i}

⟳Evaluate the differentiability condition along

h = h e_{i}

; the linear map's action on

e_{i}

recovers exactly

\partial f / \partial x_{i}

. The converse is FALSE — see the counterexample below.

Why It Holds / Derivation

Clairaut's Theorem via the Mean Value Theorem

We prove [3] for $f (x, y)$ near $(a, b)$ .

Step 1. Define the second difference

Δ (h, k) = f (a + h, b + k) - f (a + h, b) - f (a, b + k) + f (a, b) .

Step 2. Set

φ (x) = f (x, b + k) - f (x, b)

, so

Δ = φ (a + h) - φ (a)

. By the Mean Value Theorem there is

ξ

between

a

and

a + h

with

Δ = h φ^{'} (ξ) = h [f_{x} (ξ, b + k) - f_{x} (ξ, b)] .

Step 3. Apply the MVT again to

y \mapsto f_{x} (ξ, y)

[b, b + k]

: there is

η

between

b

and

b + k

with

Δ = h k f_{y x} (ξ, η) .

Step 4. By the symmetric argument (subtracting in the other order, defining

ψ (y) = f (a + h, y) - f (a, y)

), we obtain

Δ = h k f_{x y} (ξ^{'}, η^{'})

for some

(ξ^{'}, η^{'})

. Step 5. Equate:

f_{y x} (ξ, η) = f_{x y} (ξ^{'}, η^{'})

. As

h, k \to 0

, all four sample points tend to

(a, b)

. Continuity of both mixed partials forces

f_{y x} (a, b) = f_{x y} (a, b)

■

Carl Gustav Jacob Jacobi | 1804–1851

Jacobi systematized the use of partial derivatives in his work on determinants of derivative arrays — what we now call the Jacobian matrix and Jacobian determinant. His 1841 treatise De determinantibus functionalibus fixed the modern $\partial$ notation in the mathematical mainstream and demonstrated the power of organizing partial derivatives into matrices for studying coordinate transformations, implicit functions, and systems of equations. The chain rule for compositions of multivariable maps is most naturally stated through Jacobian matrix multiplication.

Building Intuition

Imagine a mountainous terrain whose height is $z = f (x, y)$ , with $x$ pointing east and $y$ pointing north. Standing at a point, $\partial f / \partial x$ is the slope you feel if you walk due east — you ignore any north–south change because $y$ is held fixed. Likewise $\partial f / \partial y$ is the slope walking due north. Each partial is a one-dimensional slice: cut the surface with a vertical plane aligned to an axis, and you get an ordinary curve whose slope is the partial derivative. The full gradient $\nabla f = (\partial f / \partial x, \partial f / \partial y)$ assembles these slices into the single direction of steepest ascent.

Slicing a surface: the partial derivative ∂f/∂x is the slope of the curve cut by the plane y = b

Worked Examples

Polynomial Partials with Numbers

Given:

f (x, y) = 3 x^{2} y + 4 x y^{3} - 5 y + 7

. Find

\partial f / \partial x

and

\partial f / \partial y

at the point

(2, 1)

. Step 1 — Differentiate w.r.t. $x$ (treat $y$ as constant):

\frac{\partial f}{\partial x} = 6 x y + 4 y^{3} .

Step 2 — Evaluate at $(2, 1)$ :

\frac{\partial f}{\partial x} (2, 1) = 6 (2) (1) + 4 (1)^{3} = 12 + 4 = 16.

Step 3 — Differentiate w.r.t. $y$ (treat $x$ as constant):

\frac{\partial f}{\partial y} = 3 x^{2} + 12 x y^{2} - 5.

Step 4 — Evaluate at $(2, 1)$ :

\frac{\partial f}{\partial y} (2, 1) = 3 (4) + 12 (2) (1) - 5 = 12 + 24 - 5 = 31.

Boxed answer:

f_{x} (2, 1) = 16, f_{y} (2, 1) = 31

(dimensionless slopes; if

x, y

carry units, slopes carry units of

f

per unit of the respective variable).

Ideal Gas — Partial Derivatives in Thermodynamics

Given: The ideal gas law

P = \frac{n R T}{V}

, with

n = 2 mol

R = 8.314 J\cdotmol^{- 1} K^{- 1}

T = 300 K

V = 0.05 m^{3}

. Find how pressure changes with temperature (at fixed

V

) and with volume (at fixed

T

). Step 1 — Partial w.r.t. $T$ at constant $V$ :

(\frac{\partial P}{\partial T})_{V} = \frac{n R}{V} = \frac{( 2 ) ( 8.314 )}{0.05} = 332.6 Pa\cdotK^{- 1} .

Step 2 — Partial w.r.t. $V$ at constant $T$ :

(\frac{\partial P}{\partial V})_{T} = - \frac{n R T}{V ^{2}} = - \frac{( 2 ) ( 8.314 ) ( 300 )}{( 0.05 ) ^{2}} = - 1.996 \times 1 0^{6} Pa\cdotm^{- 3} .

Boxed answer:

(\partial P / \partial T)_{V} = + 332.6 Pa/K, (\partial P / \partial V)_{T} = - 2.00 \times 1 0^{6} Pa/m^{3}

. The subscripts

V

and

T

explicitly state what is held constant — essential in thermodynamics.

Common Misconceptions

The subscript "held constant" is not optional in physics. Writing

\partial P / \partial T

is ambiguous in thermodynamics because

P

can be expressed in terms of different variable sets. The notation

(\partial P / \partial T)_{V}

versus

(\partial P / \partial T)_{S}

(constant volume vs. constant entropy) gives genuinely different numbers. Always declare which variables are frozen. Mixed partials are NOT always equal. The interchange

f_{x y} = f_{y x}

requires continuity of the mixed second derivatives (Clairaut's hypothesis). Without it, the order of differentiation can matter — see the counterexample.

Existence of Partials Does Not Imply Continuity

Consider

f (x, y) = ⎩ ⎨ ⎧ \frac{x y}{x ^{2} + y ^{2}}, 0, (x, y) \neq = (0, 0) (x, y) = (0, 0) .

At the origin both partials exist: along the

x

-axis

f (x, 0) = 0

, so

f_{x} (0, 0) = 0

; similarly

f_{y} (0, 0) = 0

. Yet

f

is not continuous at

(0, 0)

— along the line

y = x

f (x, x) = \frac{1}{2}

for all

x \neq = 0

, so the limit is

\frac{1}{2} \neq = 0 = f (0, 0)

. Thus a function can have all first-order partials at a point and still be discontinuous there. Partial differentiability is strictly weaker than total differentiability.

Mixed Partials That Disagree

For

g (x, y) = ⎩ ⎨ ⎧ \frac{x y ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}}, 0, (x, y) \neq = (0, 0) (x, y) = (0, 0),

one computes

g_{x y} (0, 0) = - 1

but

g_{y x} (0, 0) = + 1

. The mixed second partials exist but are unequal because they fail to be continuous at the origin — exactly the situation Clairaut's theorem excludes.

Connections

Partial derivatives are the components of the Gradient, whose direction gives steepest ascent and whose magnitude gives the maximal rate of change. Assembling partials of a vector-valued map produces the Jacobian Matrix, which governs the multivariable Chain Rule and the Inverse Function Theorem. When set to zero, partials locate stationary points in Multivariable Optimization, with the Hessian Matrix of second partials classifying them. Partial derivatives are the alphabet of Partial Differential Equations, and the Total Differential in [4] links them to Differentiability in Several Variables. They also underlie the Directional Derivative, of which they are the axis-aligned special case.

Real-World Applications

Thermodynamics (Maxwell relations). Equalities like $(\partial T / \partial V)_{S} = - (\partial P / \partial S)_{V}$ derive directly from equality of mixed partials of thermodynamic potentials.
Machine learning (backpropagation). Training neural networks computes partial derivatives of a loss function with respect to millions of weights; gradient descent uses these to update parameters.
Economics (marginal analysis). Marginal utility $\partial U / \partial x_{i}$ and marginal product $\partial Q / \partial L$ (holding capital fixed) quantify the effect of one input.
Fluid dynamics (Navier–Stokes equations). Velocity fields evolve via PDEs built from spatial and temporal partial derivatives.
Electromagnetism (Maxwell's equations). Divergence and curl are combinations of partial derivatives of the field components.
Heat transfer (the heat equation). $\partial u / \partial t = α \nabla^{2} u$ relates the time partial to the sum of second spatial partials.
Quantitative finance (the Greeks). The option sensitivities Delta $= \partial V / \partial S$ , Theta $= \partial V / \partial t$ , and Vega $= \partial V / \partial σ$ are all partial derivatives of an option's price.

Historical Development

1690s — Leibniz and the Bernoullis differentiate expressions in several variables informally, treating extra variables as parameters.
1734 — Leonhard Euler develops systematic methods for functions of several variables and conditions for exactness of differentials.
1740s — Alexis Clairaut and Euler independently establish the symmetry of mixed second derivatives for smooth functions.
1770s — Lagrange advances the theory in the calculus of variations, treating partial derivatives of action-like functionals.
1786 — Adrien-Marie Legendre introduces a rounded " $\partial$ " symbol to distinguish partial from total differentiation.
1841 — Jacobi's De determinantibus functionalibus firmly standardizes the $\partial$ notation and the use of derivative matrices (Jacobians).
1860s–1870s — Weierstrass and others place limits and differentiability on rigorous $ε$ – $δ$ foundations, clarifying the gap between partial and total differentiability.
1880s — Hermann Schwarz gives the careful modern statement and proof of the equality-of-mixed-partials theorem (hence "Clairaut–Schwarz").
20th century — The notion generalizes to the Fréchet and Gâteaux derivatives in infinite-dimensional spaces, and partial derivatives become central to the theory of distributions and PDEs.

Subtleties & Limits

The frontier is not whether partials exist, but how weakly differentiable a function can be while still admitting meaningful "partial derivatives." Sobolev spaces replace classical partials with weak derivatives defined through integration by parts, allowing differential equations to have solutions that are not differentiable in the classical sense. Determining the optimal regularity of solutions to nonlinear PDEs — for instance, whether solutions to the 3D incompressible Navier–Stokes equations stay smooth (a Clay Millennium Problem) — hinges on controlling partial derivatives. Even at the undergraduate boundary, the precise gap between "all partials exist," "all directional derivatives exist," and "fully differentiable" reveals how subtle multivariable smoothness truly is.