Partial Derivative Symbol (∂): Meaning, Uses and How to Type It

The partial derivative symbol ∂ is a rounded, curly d. It marks a derivative taken with respect to one variable while every other variable is held fixed: for f(x, y) = x²y, ∂f/∂x = 2xy. You will find it in multivariable calculus, in physics equations such as the heat and wave equations, and in the gradients that train machine-learning models.

Below: the character to copy, the codes for every platform, how ∂ is used, and how to write partial derivatives in LaTeX.

Partial Derivative Symbol (∂): Copy and Paste

Copy the symbol from the box below, or take the code you need from the table:

1∂
Copy

∂ is not the Greek letter delta δ, and not the Cyrillic letter д. Use the real character so formulas stay searchable.

Partial Derivative Definition

For a function of several variables, the partial derivative with respect to x is the ordinary derivative you get by treating all other variables as constants:

∂f/∂x = lim (h → 0) [f(x + h, y) − f(x, y)] / h

Higher derivatives stack the symbol: ∂²f/∂x² differentiates twice in x, and the mixed partial ∂²f/∂x∂y differentiates once in each variable. For smooth functions the order does not matter (Schwarz's theorem). A common shorthand writes the variable as a subscript, fₓ or ∂ₓf.

The rounded d was used by Adrien-Marie Legendre in 1786, abandoned, and reintroduced by Carl Gustav Jacob Jacobi in 1841, whose usage stuck.

Uses of the Partial Derivative Symbol

• Partial differential equations: the heat equation ∂u/∂t = α ∂²u/∂x², the wave equation ∂²u/∂t² = c² ∂²u/∂x², and the Schrödinger equation iℏ ∂ψ/∂t = Ĥψ all use ∂ because u and ψ depend on both space and time.
• Gradients and Jacobians: the gradient collects partial derivatives, ∇f = (∂f/∂x, ∂f/∂y), and backpropagation computes ∂L/∂w for every weight w with the chain rule. See the nabla symbol.
• Thermodynamics: subscripts show which variables are held constant, as in (∂U/∂S)_V = T. The same derivative with a different variable held fixed has a different value.
• Topology and geometry: ∂A denotes the boundary of a set or manifold A, as in Stokes' theorem ∫_∂Ω ω = ∫_Ω dω.

How to Type the Partial Derivative Symbol

• Mac: press Option + D on the US and UK layouts. The Character Viewer (search "partial") works on any layout.
• Windows: there is no Alt code. In Word, type 2202 and press Alt + X, or insert an equation (Alt + =) and type \partial followed by a space.
• Google Docs: Insert → Special characters and search "partial," or type \partial inside an equation.
• HTML: ∂ or ∂.
• Phones: neither iOS nor Gboard has ∂; copy it or add a text replacement.

Partial Derivative in LaTeX

Write \partial in math mode and build the fraction with \frac. The physics package shortens this to \pdv, but plain LaTeX is shown first:

1\documentclass{article}
2\usepackage{amsmath}
3\begin{document}
4First and second partials:
5\[ \frac{\partial f}{\partial x}, \qquad
6   \frac{\partial^2 f}{\partial x^2}, \qquad
7   \frac{\partial^2 f}{\partial x \, \partial y} \]
8
9Heat equation: $\dfrac{\partial u}{\partial t} = \alpha \dfrac{\partial^2 u}{\partial x^2}$
10
11Held constant: $\left( \frac{\partial U}{\partial S} \right)_{V} = T$
12
13Shorthand and boundary: $\partial_x f$, \quad $\partial \Omega$
14\end{document}
Copy

• With \usepackage{physics}, \pdv{f}{x} gives ∂f/∂x, \pdv[2]{f}{x} the second derivative and \pdv{f}{x}{y} the mixed partial.
• Inline, \frac shrinks the fraction; use \dfrac (amsmath) or \partial f / \partial x when it must stay readable.
• Add a thin space \, between \partial x and \partial y in mixed partials.

For the derivative rules themselves, see the derivative rules guide. Every calculus symbol with its command is on the calculus symbol list, and the Greek letters that look similar are compared in the delta symbols guide.

Related symbols and guides

• Partial Derivative symbol page — copy the symbol and its LaTeX command
• Calculus symbols reference — every symbol in the category with its LaTeX command
• Derivative rules — how to typeset these symbols in a document
• Nabla symbol (∇)
• Delta in math: change and difference

Partial Derivative Symbol FAQ

What does ∂ mean in math?

∂ marks a partial derivative: the rate of change of a multivariable function with respect to one variable while the others are held constant. In topology, ∂A also means the boundary of A.

How do you pronounce ∂?

"Partial," "del," "dee" or "curly d." ∂f/∂x is usually read "partial f partial x" or "the partial of f with respect to x."

What is the difference between d and ∂?

d is used for the derivative of a function of one variable, df/dx. ∂ is used when the function depends on several variables and only one is varied, ∂f/∂x.

How do I type ∂ in LaTeX?

Use \partial in math mode, for example $\frac{\partial f}{\partial x}$. With the physics package you can write \pdv{f}{x} instead.

How do I type ∂ on a keyboard?

On a Mac press Option + D. On Windows there is no Alt code; in Word type 2202 then Alt + X, or copy the symbol from this page.