Partial Derivative Notation
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 4.3: “Partial Derivatives” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives |
| Textbook used in class | Stewart, Calculus, Section 14.3 (notation used throughout) |
Quick Reference
| Subscript | Leibniz | Meaning |
|---|---|---|
| $f_x$ | $\partial f/\partial x$ | differentiate $f$ once with respect to $x$ |
| $f_y$ | $\partial f/\partial y$ | differentiate $f$ once with respect to $y$ |
| $f_{xy}$ | $\partial^2 f/\partial y\,\partial x$ | differentiate first w.r.t. $x$, then w.r.t. $y$ |
| $f_{yx}$ | $\partial^2 f/\partial x\,\partial y$ | differentiate first w.r.t. $y$, then w.r.t. $x$ |
Order in subscript notation: $f_{xy}$ means “first $x$, then $y$” -- left to right.
Order in Leibniz notation: $\partial^2 f/\partial y\,\partial x$ means “first $x$ (innermost), then $y$ (outermost)” -- right to left.
Motivation
Partial derivatives can be written in two main notations, and reading them correctly is essential because textbooks and problems use both. The subscript notation $f_{xy}$ is compact; the Leibniz notation $\partial^2 f/\partial y\,\partial x$ makes the order of operations explicit. The tricky part: the two notations stack derivatives in opposite orders, which catches nearly every student at least once.
Key Concepts
1. First-Order Notation
For $z = f(x,y)$:
- “Partial of $f$ with respect to $x$”: written $f_x$, $\partial f/\partial x$, $\partial z/\partial x$, or $D_x f$.
- “Partial of $f$ with respect to $y$”: written $f_y$, $\partial f/\partial y$, $\partial z/\partial y$, or $D_y f$.
All of these mean the same thing: differentiate once, holding the other variable constant.
2. Mixed Second-Order Notation
$f_{xy}$ (subscript): differentiate with respect to $x$ FIRST, then with respect to $y$.
Written in steps: $f_{xy} = (f_x)_y = \dfrac{\partial}{\partial y}\!\left(\dfrac{\partial f}{\partial x}\right) = \dfrac{\partial^2 f}{\partial y\,\partial x}$.
Notice that in the Leibniz form $\partial^2 f / \partial y\,\partial x$, the variable closest to $f$ (innermost) is differentiated first. This is the OPPOSITE order from subscript notation.
Memory trick: In subscripts, read left to right. In Leibniz, read right to left (innermost first, like function composition).
Worked Example
Let $f(x,y) = x^2 y^3$.
$f_x$: treat $y$ as constant: $f_x = 2xy^3$.
$f_y$: treat $x$ as constant: $f_y = 3x^2y^2$.
$f_{xy}$: differentiate $f_x = 2xy^3$ with respect to $y$: $f_{xy} = 6xy^2$.
$f_{yx}$: differentiate $f_y = 3x^2y^2$ with respect to $x$: $f_{yx} = 6xy^2$.
In this case $f_{xy} = f_{yx}$. Clairaut’s theorem (see the next skill) guarantees this equality whenever the mixed partials are continuous.
In Leibniz notation: $f_{xy} = \dfrac{\partial^2 f}{\partial y\,\partial x}$ and $f_{yx} = \dfrac{\partial^2 f}{\partial x\,\partial y}$.
$\partial^2 f/\partial y\,\partial x$ means “differentiate with respect to $y$ first.” The Leibniz notation $\partial^2 f/\partial y\,\partial x$ means differentiate with respect to $x$ first (the variable written CLOSEST to $f$), then with respect to $y$. This is because you read the operator $\dfrac{\partial}{\partial y}\dfrac{\partial}{\partial x}$ right-to-left, applying the innermost operator first -- just like function composition. The subscript notation $f_{xy}$ reads left-to-right. These two notations describe the SAME operation, but they stack the variables in opposite orders.
Leveled Practice
Problem 1. Compute $\dfrac{\partial^2 f}{\partial x\,\partial y}$ and $\dfrac{\partial^2 f}{\partial y\,\partial x}$ for $f(x,y) = x\sin(y) + y e^x$.
Show answer
$\dfrac{\partial^2 f}{\partial x\,\partial y}$ means $f_{yx}$: first differentiate w.r.t. $y$, then w.r.t. $x$.
$f_y = x\cos(y) + e^x$. Then $f_{yx} = \cos(y) + e^x$.
$\dfrac{\partial^2 f}{\partial y\,\partial x}$ means $f_{xy}$: first differentiate w.r.t. $x$, then w.r.t. $y$.
$f_x = \sin(y) + ye^x$. Then $f_{xy} = \cos(y) + e^x$.
They are equal (as expected by Clairaut’s theorem).