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 partialderivative symbol is ∂. The notation was introduced by AdrienMarie 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 xzplane, and those that are parallel to the yzplane.
To find the slope of the line tangent to the function at (1, 1, 3) that is parallel to the xzplane, 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:
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