partial derivative — from formula anatomyA partial derivative measures how a function changes when only one of its multiple variables is varied, while all others are held constant. It is a cornerstone of multivariable calculus and appears throughout physics, from thermodynamics (where pressure, volume, and temperature are interdependent) to field theories (where fields depend on space and time). The notation ∂f/∂x denotes the partial derivative of f with respect to x.
The concept was developed in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, who extended differential calculus to functions of several variables. The modern notation ∂ was introduced by Marquis de Condorcet in 1770 and popularized by Carl Gustav Jacob Jacobi in the 19th century.
To compute a partial derivative, treat all variables except the one of interest as constants and differentiate as in ordinary calculus. For example, if f(x, y) = x²y + sin(y), then ∂f/∂x = 2xy, and ∂f/∂y = x² + cos(y). Partial derivatives are used to build gradients, divergences, and curls in vector calculus, and they appear in differential equations like the wave equation and the Schrödinger equation.
Partial derivatives are used in optimization (e.g., gradient descent in machine learning), in solving partial differential equations for fluid dynamics and electromagnetism, and in error propagation analysis in experimental physics.
In thermodynamics, the four Maxwell relations are derived from the equality of mixed partial derivatives, linking quantities like temperature, pressure, entropy, and volume.
| c_s | variable | sound speed — sound speed metre per second |
| P | variable | pressure of the scalar field — pressure pascal |
| X | variable | kinetic term, -1/2 (∂φ)^2 — kinetic energy density joule per cubic metre |
| P_X | variable | partial derivative of P with respect to X — partial derivative |
| P_XX | variable | second partial derivative of P with respect to X — second partial derivative |
| v | variable | null coordinate (e.g., advanced time) — null coordinate metre |
| r | variable | radial coordinate — radial coordinate metre |
| ∂_v | operator | differentiates with respect to the null coordinate — partial derivative |
| P | variable | Probability density — probability density per metre cubed |
| \Psi | variable | Wave function — wave function per metre cubed half |
| \Psi^* | variable | Complex conjugate of wave function — wave function conjugate per metre cubed half |
| t | variable | Time — time second |
| \frac{\partial}{\par | operator | Partial derivative with respect to time — partial derivative |
| ψ | variable | wave function of the quantum particle — wave function metre to the minus 3 over 2 |
| t | variable | time — time second |
| V | variable | potential energy (here self-gravitating) — potential energy joule |
| m | variable | mass of the particle — mass kilogram |
| ħ | constant | Reduced Planck constant, fundamental quantum of action — reduced planck constant = 1.054571817e-34 joule second |
| ∂/∂t | operator | partial derivative with respect to time — partial derivative |
| ∇² | operator | Laplacian (spatial second derivative) — laplacian operator |
| β_ρ | variable | beta function for ρ joule per cubic metre |
| M | variable | renormalization scale — energy joule |
| ρ | variable | running parameter joule per cubic metre |
| ∂/∂M | operator | partial derivative with respect to scale M — partial derivative |
| ΔT | variable | temperature change — temperature kelvin |
| T | variable | absolute temperature — temperature kelvin |
| C_p | variable | heat capacity at constant pressure — heat capacity joule per kelvin |
| S | variable | entropy — entropy joule per kelvin |
| ΔP | variable | pressure change — pressure pascal |
| ∂/∂P | operator | partial derivative with respect to pressure at constant temperature — partial derivative |