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

Consider a variable z that depends on x and y, while both x and y depend on another variable t.Although t does not appear directly in the expression for z, variation in t alters x and y, resulting in an indirect change in z. Here, the goal is to find the total derivative, dz/dt.Assuming all functions are differentiable, the total change in z is the sum of its partial changes with respect to x and y.By dividing this relationship by Delta t and taking the limit as Delta t approaches zero, the ratios transform into derivatives.This shows that the total derivative is the sum of the derivatives of x and y with respect to t, each multiplied by the corresponding partial derivative of the original function, giving the Multivariable Chain Rule.For example, consider a weather balloon whose temperature depends on both altitude and humidity. As the balloon rises, both these variables change over time.Using the chain rule for multivariable functions, the total temperature change is the sum of: how temperature varies with altitude times the balloon&#x27;s speed, plus how the temperature varies with humidity times the humidity&#x27;s rate of change over time.

multivariable chain rule

When a variable z depends on two intermediate variables, x and y, and both x and y vary with respect to a third variable t, the dependence of z on t is indirect. Although t does not explicitly appear in the expression for z, any change in t produces corresponding changes in x and y, which in turn alter the value of z. The objective is to determine the total derivative of z with respect to t, denoted as dz/dt.Assuming that all functions involved are differentiable, the total change in z can be expressed as the sum of its partial changes with respect to x and y. Each partial change measures how z responds to variation in one variable while the other is held constant. When these changes are divided by a small increment in t and the limit is taken as the increment approaches zero, the resulting ratios become derivatives.This process demonstrates that the total derivative of z with respect to t equals the derivative of x with respect to t multiplied by the partial derivative of z with respect to x, plus the derivative of y with respect to t multiplied by the partial derivative of z with respect to y. This relationship is known as the multivariable chain rule. It provides a systematic method for computing rates of change when variables are interconnected through intermediate dependencies.A practical illustration is a weather balloon whose temperature depends on altitude and humidity. As the balloon ascends, both altitude and humidity vary over time. The total rate of change of temperature is therefore the sum of two contributions: the rate at which temperature changes with altitude multiplied by the balloon&#x2019;s vertical speed, and the rate at which temperature changes with humidity multiplied by the rate at which humidity changes over time.

Multivariable Chain Rule

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Education

JoVE Core

Calculus

Unit

Partial Derivatives and Gradients

KEY TERMS AND CONCEPTS

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

functions of two variables

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tangent planes to surfaces

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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...

implicit differentiation with partial derivatives

implicit-differentiation-with-partial-derivatives

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key terms and concepts

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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