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			File:Partial_func_eg.svg\|Graph of a function with partial derivatives
			File:X2+X+1.svg\|Graph of the quadratic function x^2 + x + 1
			File:Cone_3d.png\|3D model of a cone
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2024-03-25T02:15:07Z

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The '''partial derivative''' is a fundamental concept in the field of [[calculus]] that extends the idea of a [[derivative]] to functions of multiple variables. It represents the rate at which a function changes as one of its variables is varied, while the other variables are held constant. Partial derivatives are crucial in various fields such as [[physics]], [[engineering]], and [[economics]], where they are used to study the behavior of physical systems, design complex structures, and analyze economic models, respectively.

==Definition==
Given a function \(f(x_1, x_2, ..., x_n)\) of several variables, the partial derivative of \(f\) with respect to the variable \(x_i\) is denoted as \(\frac{\partial f}{\partial x_i}\). It is defined as the limit:

\[
\frac{\partial f}{\partial x_i} = \lim_{\Delta x_i \to 0} \frac{f(x_1, ..., x_i + \Delta x_i, ..., x_n) - f(x_1, ..., x_i, ..., x_n)}{\Delta x_i}
\]

This definition mirrors that of the ordinary derivative, but it applies to functions of more than one variable.

==Applications==
Partial derivatives play a key role in various applications:

* In [[physics]], they are used to formulate the laws of nature in the language of [[differential equations]]. For example, the [[Maxwell's equations]] that describe how electric and magnetic fields propagate.
* In [[engineering]], partial derivatives are used in the design and analysis of systems. For instance, in [[thermodynamics]], they help describe how the state of a system changes in response to changes in its properties, such as volume or pressure.
* In [[economics]], partial derivatives are used to model how different factors affect the outcome of economic models. For example, they can describe how changing the price of a good affects demand.

==Higher-Order Partial Derivatives==
Partial derivatives can themselves be differentiated with respect to another variable, leading to higher-order partial derivatives. The notation for the second-order partial derivative of \(f\) with respect to \(x_i\) and then \(x_j\) is \(\frac{\partial^2 f}{\partial x_j \partial x_i}\). These higher-order derivatives can provide deeper insights into the behavior of functions, especially in the study of [[optimization]] and [[curvature]] of surfaces.

==Mixed Partial Derivatives==
A mixed partial derivative is a second-order partial derivative where the differentiation is performed with respect to two different variables. The [[Clairaut's theorem]] on equality of mixed partials states that if \(f\) is a function of two variables that is continuously differentiable, then the mixed partial derivatives are equal, that is, \(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\).

==See Also==
* [[Derivative]]
* [[Differential equation]]
* [[Gradient]]
* [[Jacobian matrix and determinant]]
* [[Laplace operator]]

[[Category:Calculus]]
[[Category:Mathematical analysis]]
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Partial derivative - Revision history

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