partial derivative

简明释义

偏导数

英英释义

例句

1.To find the maximum profit, we need to calculate the partial derivative 偏导数 of the profit function with respect to each variable.

为了找到最大利润，我们需要计算利润函数关于每个变量的偏导数 partial derivative。

2.In economics, the partial derivative 偏导数 of a utility function helps determine how much satisfaction increases with an additional unit of a good.

在经济学中，效用函数的偏导数 partial derivative有助于确定增加一单位商品带来的满足感提升多少。

3.When calculating the rate of change of temperature with respect to time, we often use the partial derivative 偏导数 of the temperature function.

在计算温度随时间变化的速率时，我们通常使用温度函数的偏导数 partial derivative。

4.The partial derivative 偏导数 with respect to x in a multivariable function shows the slope of the function in the x-direction.

多变量函数中关于x的偏导数 partial derivative显示了函数在x方向的斜率。

5.In physics, the partial derivative 偏导数 of a function can describe how a physical quantity changes with respect to one variable while keeping others constant.

在物理学中，函数的偏导数 partial derivative可以描述一个物理量在保持其他变量不变的情况下如何随一个变量变化。

作文

In the field of mathematics, particularly in calculus, the concept of partial derivative is fundamental for understanding how functions behave with respect to multiple variables. A partial derivative refers to the derivative of a function taken with respect to one variable while holding the other variables constant. This allows us to analyze the effect of changing one variable at a time, which is crucial for multivariable functions where interactions between different variables can complicate analysis.For example, consider a function that describes the temperature in a room as a function of both time and the position within the room. If we want to understand how the temperature changes over time at a specific location, we would take the partial derivative of the temperature function with respect to time. This provides insight into how quickly the temperature is rising or falling at that particular spot without considering how the position affects the temperature.The notation for partial derivatives typically involves the symbol ∂. For instance, if we have a function f(x, y), the partial derivative of f with respect to x is denoted as ∂f/∂x, while the partial derivative with respect to y is denoted as ∂f/∂y. These notations signify that we are focusing on how the output of the function changes when we vary either x or y independently.Understanding partial derivatives is essential in various fields, including physics, engineering, economics, and data science. In physics, for instance, the concept is used to describe how physical quantities change in space and time. Engineers utilize partial derivatives in optimization problems, where they need to find the best solution under certain constraints. Economists often use partial derivatives to evaluate how changes in one economic variable, such as price, affect another variable, like quantity demanded, while keeping other factors constant.Moreover, the application of partial derivatives extends to machine learning and artificial intelligence. When training models, we often deal with functions that depend on numerous parameters. By calculating the partial derivatives of a loss function with respect to each parameter, we can determine how to adjust those parameters to minimize the loss, thereby improving the model's performance.In summary, the concept of partial derivative is a powerful tool in mathematics and its applications. It allows for a clearer understanding of complex functions involving multiple variables by isolating the effects of individual variables. Mastering partial derivatives equips students and professionals alike with the skills necessary to tackle real-world problems across various disciplines. As we delve deeper into the world of mathematics and its applications, the importance of partial derivatives cannot be overstated; they are indeed a cornerstone of multivariable calculus and its many applications in science and engineering.

在数学领域，特别是在微积分中，偏导数的概念对于理解多变量函数的行为至关重要。偏导数是指在保持其他变量不变的情况下，对一个变量进行求导。这使我们能够逐一分析改变一个变量的影响，这对于多变量函数来说非常重要，因为不同变量之间的相互作用可能会使分析变得复杂。例如，考虑一个描述房间内温度随时间和位置变化的函数。如果我们想了解特定位置的温度如何随时间变化，我们将对温度函数关于时间进行偏导数运算。这使我们能够了解在特定位置温度上升或下降的速度，而不考虑位置对温度的影响。偏导数的符号通常使用∂符号。例如，如果我们有一个函数f(x, y)，那么f关于x的偏导数表示为∂f/∂x，而关于y的偏导数表示为∂f/∂y。这些符号表明我们专注于当我们独立改变x或y时，函数输出如何变化。理解偏导数在物理、工程、经济学和数据科学等多个领域中至关重要。在物理学中，这个概念用于描述物理量在空间和时间中的变化。工程师在优化问题中利用偏导数，他们需要在某些约束条件下找到最佳解决方案。经济学家经常使用偏导数来评估一个经济变量（如价格）的变化如何影响另一个变量（如需求量），同时保持其他因素不变。此外，偏导数的应用还扩展到机器学习和人工智能领域。在训练模型时，我们通常处理依赖于多个参数的函数。通过计算损失函数对每个参数的偏导数，我们可以确定如何调整这些参数以最小化损失，从而提高模型的性能。总之，偏导数的概念是数学及其应用中的一种强大工具。它通过孤立各个变量的影响，使我们更清晰地理解涉及多个变量的复杂函数。掌握偏导数使学生和专业人员能够应对各个学科中的现实问题。当我们深入探讨数学及其应用的世界时，偏导数的重要性不容低估；它们确实是多变量微积分及其在科学和工程中众多应用的基石。