partial derivative

简明释义

偏导数

英英释义

例句

1.The partial derivative 偏导数 of temperature with respect to pressure is crucial in thermodynamics.

温度关于压力的偏导数 偏导数在热力学中至关重要。

2.When optimizing a function, we set the partial derivatives 偏导数 to zero to find critical points.

在优化一个函数时，我们将偏导数 偏导数设置为零，以找到临界点。

3.The partial derivative 偏导数 with respect to x gives us the slope of the tangent line in the x-direction.

关于x的偏导数 偏导数给出了切线在x方向上的斜率。

4.In multivariable calculus, we often calculate the partial derivative 偏导数 of a function to analyze its behavior with respect to one variable while keeping others constant.

在多变量微积分中，我们经常计算一个函数的偏导数 偏导数，以分析它在一个变量变化时的行为，同时保持其他变量不变。

5.In machine learning, understanding the partial derivatives 偏导数 of loss functions helps improve model accuracy.

在机器学习中，理解损失函数的偏导数 偏导数有助于提高模型的准确性。

作文

In the field of mathematics, particularly in calculus, the concept of partial derivative plays a crucial role in understanding how functions behave when they depend on multiple variables. A partial derivative is defined as the derivative of a function with respect to one variable while keeping the other variables constant. This allows mathematicians and scientists to analyze the impact of changing one specific variable on the overall function, which is essential in various applications such as physics, engineering, and economics.To illustrate the importance of partial derivatives, consider a function f(x, y) that represents the temperature at any point (x, y) in a two-dimensional space. If we want to understand how the temperature changes with respect to the x-coordinate while keeping the y-coordinate constant, we would compute the partial derivative of f with respect to x, denoted as ∂f/∂x. This partial derivative gives us the rate at which the temperature changes as we move along the x-axis, providing valuable insights into the behavior of the temperature distribution.Similarly, we can find the partial derivative with respect to y, denoted as ∂f/∂y, to analyze how temperature changes as we move along the y-axis. By examining both partial derivatives, we can gain a comprehensive understanding of how temperature varies in the entire two-dimensional space.The significance of partial derivatives extends beyond simple functions. In multivariable calculus, we encounter functions that depend on several variables, and partial derivatives become essential tools for optimization problems. For instance, in economics, when studying a production function that depends on labor and capital, we can use partial derivatives to determine the marginal product of labor and capital. This information helps businesses make informed decisions about resource allocation and production efficiency.Furthermore, partial derivatives are fundamental in the formulation of differential equations that describe various physical phenomena, such as fluid dynamics and heat transfer. Understanding how quantities change with respect to multiple variables allows scientists to create accurate models that predict real-world behavior.In addition to their practical applications, partial derivatives also have theoretical significance. They form the basis for more advanced topics in mathematics, such as Taylor series expansions and gradient vectors. The gradient vector, which consists of all partial derivatives of a function, provides a powerful tool for finding direction and rate of steepest ascent in multivariable functions. This concept is widely used in optimization algorithms, including machine learning techniques, where finding minima or maxima of complex functions is crucial.In conclusion, the concept of partial derivative is vital for understanding the behavior of multivariable functions. By isolating the effect of one variable while keeping others constant, partial derivatives allow us to analyze complex systems in mathematics, science, and engineering. Whether it’s determining how temperature changes in a given space or optimizing production in an economic model, partial derivatives provide essential insights that drive decision-making and innovation. Mastering this concept is not only important for academic pursuits but also for practical applications that shape our world.

在数学领域，特别是在微积分中，偏导数的概念在理解多变量函数的行为时起着至关重要的作用。偏导数被定义为一个函数对一个变量的导数，同时保持其他变量不变。这使得数学家和科学家能够分析改变一个特定变量对整体函数的影响，这在物理、工程和经济等各种应用中都是必不可少的。为了说明偏导数的重要性，考虑一个函数f(x, y)，它表示二维空间中任意点(x, y)的温度。如果我们想了解在保持y坐标不变的情况下，温度如何随x坐标的变化而变化，我们将计算f关于x的偏导数，记作∂f/∂x。这个偏导数给出了温度沿x轴变化的速率，为我们提供了温度分布行为的宝贵见解。同样，我们可以找到关于y的偏导数，记作∂f/∂y，以分析温度在沿y轴移动时如何变化。通过检查这两个偏导数，我们可以全面了解温度在整个二维空间中的变化。偏导数的重要性超出了简单函数的范围。在多变量微积分中，我们遇到依赖于多个变量的函数，而偏导数成为优化问题的重要工具。例如，在经济学中，当研究依赖于劳动和资本的生产函数时，我们可以使用偏导数来确定劳动和资本的边际产出。这些信息帮助企业做出关于资源分配和生产效率的明智决策。此外，偏导数在描述各种物理现象的微分方程的形成中也具有基础性意义，例如流体动力学和热传递。理解量如何相对于多个变量变化，使科学家能够创建准确的模型来预测现实世界的行为。除了它们的实际应用外，偏导数在理论上也具有重要意义。它们构成了数学中更高级主题的基础，例如泰勒级数展开和梯度向量。梯度向量由函数的所有偏导数组成，提供了一种强大的工具，用于寻找多变量函数中陡峭上升的方向和速率。这个概念广泛应用于优化算法，包括机器学习技术，其中找到复杂函数的最小值或最大值至关重要。总之，偏导数的概念对于理解多变量函数的行为至关重要。通过在保持其他变量不变的情况下孤立一个变量的影响，偏导数使我们能够分析数学、科学和工程中的复杂系统。无论是确定给定空间中温度的变化，还是在经济模型中优化生产，偏导数提供了推动决策和创新的基本见解。掌握这一概念不仅对学术追求重要，而且对塑造我们世界的实际应用也至关重要。