Partial derivative

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

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  Graph of a function with partial derivatives

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  Graph of the quadratic function x^2 + x + 1

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  3D model of a cone