Name: total derivatives in connected variables
Uploaded: 2026-06-08T09:38:27.930Z
Duration: 1 min 29 s
Channel: JoVE

考虑一个变量 z，它依赖于 x 和 y，而 x 和 y 均依赖于另一个变量 t。尽管 t 不直接出现在 z 的表达式中，t 的变化会改变 x 和 y，从而导致 z 发生间接变化。此处的目标是求 z 对 t 的全导数，即 dz/dt。假设所有函数均可微，则 z 的总变化量等于其对 x 和 y 的偏变化量之和。将此关系式除以 Delta t 并取 Delta t 趋近于零时的极限，这些比值便转化为导数。这表明总导数是 x 和 y 对 t 的导数之和，每个导数乘以原函数相应的偏导数，由此得到多变量链式法则。例如，考虑一个气象气球，其温度取决于海拔和湿度。随着气球上升，这两个变量均随时间变化。利用多元函数的链式法则，总温度变化等于：温度随高度的变化率乘以气球的速度，加上温度随湿度的变化率乘以湿度随时间的变化率。

多变量链式法则

当变量 z 依赖于两个中间变量时， x 和 y，以及两者 x 和 y 根据第三个变量而变化 t，依赖于 z 开启 t 是间接的。尽管 t 在表达式中未明确出现 z，任何变化 t 产生相应的变化 x 和 y，进而改变其数值 z目的是确定全导数 z 关于 t，表示为 dz/dt.假设所有涉及的函数均可微，则z的总变化量可表示为它对x和y的偏变化量之和。每个偏变化量衡量的是当另一个变量保持不变时，z对其中一个变量变化的响应程度。当这些变化量除以t的一个微小增量，并取该增量趋近于零时的极限，所得比值即为导数。该过程表明，函数 z 对 t 的全导数等于 x 对 t 的导数乘以 z 对 x 的偏导数，加上 y 对 t 的导数乘以 z 对 y 的偏导数。这一关系称为多变量链式法则。它为在变量通过中间依赖关系相互关联时计算变化率提供了一种系统的方法。一个实际的例子是气象气球，其温度取决于海拔高度和湿度。当气球上升时，海拔高度和湿度均随时间变化。因此，温度的总变化率由两部分组成：温度随海拔变化的速率乘以气球的垂直速度，加上温度随湿度变化的速率乘以湿度随时间变化的速率。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Partial Derivatives and Gradients

偏导数与梯度

KEY TERMS AND CONCEPTS

关键术语与概念

Multivariable Chain Rule

多变量链式规则

total derivatives in connected variables

Total derivatives in connected variables show how one change can affect several related quantities. Here, z depends on two intermediate variables, x and y, and both x and y change with a third variable, t. The connection is indirect, because t does not appear in the formula for z itself.

The goal is to find the total derivative of z with respect to t, written as dz/dt. When the functions are differentiable, the change in z can be split into two parts. One part comes from how z changes with x while y stays fixed. The other part comes from how z changes with y while x stays fixed.

After dividing these changes by a small change in t and taking the limit as that change approaches zero, the result is a chain rule formula. It says that dz/dt equals dx/dt multiplied by the partial derivative of z with respect to x, plus dy/dt multiplied by the partial derivative of z with respect to y. This method gives a clear way to compute rates of change when variables depend on each other through intermediate steps.

A weather balloon gives a real-world example. Its temperature depends on altitude and humidity. As the balloon rises, altitude and humidity both change over time. So the total rate of change in temperature has two parts: the effect of altitude times the balloon's vertical speed, and the effect of humidity times the rate at which humidity changes with time.

Total Derivatives in Connected Variables

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multivariable chain rule

Total derivatives in connected variables show how one change can affect several related quantities. Here, z depends on two intermediate variables, x and y, and both x and y change with a third variable, t. The connection is indirect, because t does not appear in the formula for z itself. The goal is to find the total derivative of z with respect to t, written as dz/dt. When the functions are differentiable, the change in z can be split into two parts. One part comes from how z changes with x...

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Learn how the total derivative tracks change through linked variables, using partial derivatives and a weather balloon example for grades 9–12.

Total Derivatives in Connected Variables | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Total Derivatives in Connected Variables

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