Partial Derivative
A partial derivative is the derivative of a multivariable function with respect to one variable while all other variables are held constant, measuring the function's instantaneous rate of change in a single coordinate direction. Denoted by the symbol ∂ (a stylized "d"), partial derivatives generalize single-variable differentiation to functions of two or more variables. They are essential in multivariable calculus, optimization, thermodynamics, fluid dynamics, and the formulation of partial differential equations.
Key Formula
∂f/∂x = lim(h→0) [f(x+h, y) − f(x, y)] / h
LaTeX: \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,\,y) - f(x,\,y)}{h}
| Symbol | Meaning | Unit |
|---|---|---|
| ∂f/∂x | Partial derivative of f with respect to x | depends on f |
| h | Small increment in x | dimensionless |
| y | Other variable held constant | dimensionless |
| f(x, y) | Multivariable function | dimensionless |
Worked Example
Problem
Find both partial derivatives of f(x, y) = 3x²y + y³ − 5x.
Solution
Step 1: ∂f/∂x — treat y as constant: ∂f/∂x = 6xy + 0 − 5 = 6xy − 5. Step 2: ∂f/∂y — treat x as constant: ∂f/∂y = 3x² + 3y² − 0 = 3x² + 3y².
Answer
∂f/∂x = 6xy − 5; ∂f/∂y = 3x² + 3y²
Partial Derivative Rules Compared to Single-Variable Rules
| Rule | Single-Variable | Partial Derivative (w.r.t. x) | Note |
|---|---|---|---|
| Power rule | d/dx(xⁿ) = nxⁿ⁻¹ | ∂/∂x(xⁿyᵐ) = nxⁿ⁻¹yᵐ | y treated as constant |
| Sum rule | d/dx(f+g) = f' + g' | ∂/∂x(f+g) = ∂f/∂x + ∂g/∂x | Same form |
| Product rule | d/dx(fg) = fg' + gf' | ∂/∂x(f·g) = g·∂f/∂x + f·∂g/∂x | y-only terms vanish |
| Chain rule | dy/dx = (dy/du)(du/dx) | ∂z/∂x = (∂z/∂u)(∂u/∂x) | More variables possible |
| Constant rule | d/dx(c) = 0 | ∂/∂x(g(y)) = 0 | y-only function → 0 |
Interactive Tools
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Related Terms
Gradient Vector
The gradient of a scalar function is a vector field whose components are the partial derivatives of the function with respect to each independent variable, pointing in the direction of the steepest rate of increase of the function. Denoted ∇f (nabla f), the gradient generalizes the ordinary derivative to multivariable functions. The gradient is fundamental in optimization (gradient descent), physics (conservative force fields), and machine learning for training neural networks.
Vector Divergence
Divergence is a scalar operator applied to a vector field that measures the net rate at which the vector field "spreads out" or "converges" at each point in space, representing the volumetric density of the outward flux of the field. A positive divergence indicates a source (field lines emanating outward), while negative divergence indicates a sink (field lines converging inward), and zero divergence means the field is incompressible. Divergence appears in Maxwell's equations for electromagnetism, the continuity equation in fluid mechanics, and Gauss's law.
Double Integral
A double integral is the extension of a definite integral to functions of two variables, computing the cumulative sum of a function f(x, y) over a two-dimensional region D in the xy-plane. Geometrically, the double integral of a non-negative function gives the volume under the surface z = f(x, y) above the region D. Double integrals are evaluated using iterated integration (Fubini's theorem), and are applied to compute areas, volumes, masses of planar objects, centre of mass, moments of inertia, and probabilities in joint distributions.
From Latin "partialis" (relating to a part). The curly ∂ symbol was introduced by Adrien-Marie Legendre in 1786 and popularized by Carl Jacobi in the 19th century to distinguish partial from ordinary derivatives. The concept emerged as mathematicians extended calculus to functions of several real variables.