How to Write Partial Derivative(∂x/∂t) in LaTeX?

Partial derivatives are fundamental in calculus, physics, and engineering for representing how a function changes with respect to one variable while keeping others constant.

This guide explains how to typeset partial derivatives in LaTeX, from basic symbols to advanced macros in a structured and easy-to-follow way.

Basic Command

The simplest way to display a partial derivative symbol in LaTeX is by using the \partial command. This produces the symbol ∂, which is the standard notation for partial derivatives.

It is ideal when writing mathematical formulas directly, without additional packages.

\frac{\partial{f}}{\partial{x}}

\frac

This is the LaTeX command for a fraction. It creates a numerator and a denominator separated by a horizontal bar.

\partial{f}

This represents the partial derivative operator applied to a function f. It is used to show the differentiation of f with respect to one variable.

\partial{x}

This part of the fraction denotes the variable with respect to which the function is differentiated.

\documentclass{article}
\begin{document}
  By definition, let $u=u(x,y,z,t)$. The partial derivative of $u$ with respect to $x$ is 
  \[
     \frac{\partial u}{\partial x} = \lim_{\ell \to 0}\frac{u(x+\ell,y,z,t)-u(x,y,z,t)}{\ell}.
  \]
\end{document}

Below are the different notations for presenting the partial derivative of a function f with respect to x.

Notation	Command
	f’_{x}
	\partial_{x}f
	D_{x}f
	D_{1}f
	\frac{\partial}{\partial{x}}f
	\frac{\partial{f}}{\partial{x}}

Using Physics Package

The physics package simplifies writing partial derivatives through the \pdv command.

It automatically arranges the notation neatly, supports higher orders, and even mixed derivatives.

\pdv[n]{f}{x} \quad \pdv{f}{x}{y}

\pdv

This command from the physics package typesets partial derivatives. It automatically places \partial symbols and fractions correctly.

[n]

This optional argument denotes the order of the derivative, e.g., [2] for second order.

{f}

This argument is the function or expression to be differentiated.

{x}

This specifies the variable with respect to which differentiation is done.

\documentclass{article}
\usepackage{physics}
\begin{document}
  \[
    \pdv{f}{x}, \quad \pdv[2]{f}{x}, \quad \pdv{f}{x}{y}, \quad \pdv[n]{f}{x}.
  \]
\end{document}

Using Custom Macros

You can use \newcommand to define your own shorthand macros for partial derivatives. This helps avoid repetitive typing and keeps your code cleaner.

\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

\newcommand

Used to define a custom LaTeX command. It can take optional arguments for flexibility.

{\pd}

The custom name of the command, which can be used later in the document.

{#1}, {#2}

Placeholders for arguments passed to the command, where #1 is the numerator and #2 is the denominator.

\documentclass{article}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\begin{document}
  \[
    \pd{u}{x}, \quad \pd{f}{y}.
  \]
\end{document}

Higher Order and Mixed Derivatives

You can typeset higher order derivatives by adjusting powers or stacking multiple \partial operators. For mixed derivatives, clearly show the order of differentiation.

\frac{\partial^2 f}{\partial x \partial y}

\partial^2 f

Represents the second-order partial derivative of f.

\partial x \partial y

Indicates that f is first differentiated with respect to y, then with respect to x.

\documentclass{article}
\begin{document}
  Mixed and higher order examples
  \[
    \frac{\partial^2 f}{\partial x \partial y}, \qquad
    \frac{\partial^n f}{\partial x_1 \partial x_2 \cdots \partial x_n}.
  \]
\end{document}

Examples in PDEs

Partial derivatives are common in equations like the heat, wave, and Laplace equations. You can display them in both fraction and subscript notation for clarity.

\documentclass{article}
\usepackage{amsmath}
\begin{document}

Heat equation:
  \[ \frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2} \quad \text{or} \quad u_t = c^2 u_{xx}. \]
  
Wave equation:
  \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \quad \text{or} \quad u_{tt} = c^2 u_{xx}. \]

Laplace equation:
  \[\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0. \]

\end{document}

Best Practice

For short equations and quick notes, use \frac{\partial f}{\partial x} since it is clear and needs no extra packages.

For research papers or documents with many derivatives, prefer \pdv from the physics package for readability and compactness.

Define macros if you frequently repeat derivative expressions.

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