Partial derivative

related topics
{math, number, function}
{math, energy, light}
{rate, high, increase}
{style, bgcolor, rowspan}

Power rule, Product rule, Quotient rule, Chain rule

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells
, substitution,
trigonometric substitution,
partial fractions, changing order

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

The partial derivative of a function f with respect to the variable x is variously denoted by

The partial-derivative symbol is . The notation was introduced by Adrien-Marie Legendre and gained general acceptance after its reintroduction by Carl Gustav Jacob Jacobi.[1]

Contents

Introduction

Suppose that ƒ is a function of more than one variable. For instance,

The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane.

To find the slope of the line tangent to the function at (1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. The graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point (x, y, z) is found to be:

Full article ▸

related documents
Inequality
Total order
Analytic continuation
Transcendental number
Octonion
Embedding
Orthogonality
Burnside's problem
Uniform convergence
Polymorphism in object-oriented programming
Planar graph
Binary heap
Additive category
Tangent space
Hausdorff space
Sufficiency (statistics)
Binary tree
Spectral theorem
Euler–Maclaurin formula
Bubble sort
Analytic function
Symmetric matrix
Array
Root-finding algorithm
Tree automaton
Self-organizing map
MathML
Shell sort
Universal property
Brouwer fixed point theorem