The Multivariable Chain Rule: A Comprehensive Study Guide

This guide covers the extension of the single-variable Chain Rule to functions of several variables, focusing on intermediate and independent variables, the use of tree diagrams, and the generalized chain rule formula.

Learning Objectives

After studying this chapter, you should be able to:

State the chain rules for one or two independent variables.
Construct and interpret tree diagrams to visualize dependency paths.
Differentiate between intermediate and independent variables in a composite function.
Perform implicit differentiation of functions involving several variables.
Apply the generalized chain rule to functions with $n$ independent variables and $m$ intermediate variables.

Key Terms & Glossary

Independent Variable: The fundamental inputs at the "bottom" of the dependency chain (e.g., $t$ in $x(t)$ or $u, v$ in $x(u,v)$ ).
Intermediate Variable: A variable that depends on others but is itself an input for a higher-level function (e.g., $x$ and $y$ in $z = f(x,y)$ where $x$ and $y$ are functions of $t$ ).
Composite Function: A function formed by substituting one function into another.
Partial Derivative: The derivative of a multivariable function with respect to one variable while holding others constant.
Total Derivative: The rate of change of a function with respect to an independent variable, accounting for all paths through intermediate variables.

The "Big Idea"

In single-variable calculus, the Chain Rule handles functions nested like Russian dolls: $f(g(x))$ . In multivariable calculus, the nesting becomes a web. A change in an independent variable (like time $t$ ) might affect multiple intermediate variables (like coordinates $x$ and $y$ ), which in turn both affect the final output $z$ . The multivariable Chain Rule is essentially a way to sum the contributions of change from every possible path through that web.

Formula / Concept Box

Case	Dependency	Formula
One Independent Variable	$z = f(x,y)$ , where $x=g(t), y=h(t)$	$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$
Two Independent Variables	$z = f(x,y)$ , where $x=g(u,v), y=h(u,v)$	$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$
Generalized Case	$z = f(x_1, ..., x_n)$ , $x_i = g_i(t_1, ..., t_m)$	$\frac{\partial z}{\partial t_j} = \sum_{i=1}^{n} \frac{\partial z}{\partial x_i} \frac{\partial x_i}{\partial t_j}$

[!IMPORTANT] Note the notation difference: If the final result depends on only one variable, we use $d$ (total derivative). If it depends on multiple variables, we use $\partial$ (partial derivative).

Hierarchical Outline

I. Chain Rule for One Independent Variable
- Dependency Path: $t \to (x, y) \to z$
- Summation Principle: Multiply derivatives along paths; add the paths together.
II. Chain Rule for Two Independent Variables
- Independence: $u$ and $v$ are independent; changing $u$ does not change $v$ .
- Partial Differentiation: We calculate $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v}$ separately.
III. The Generalized Chain Rule
- Dimensionality: Works for $n$ variables and $m$ parameters.
- Matrix View: Can be represented as a product of Jacobian matrices.
IV. Implicit Differentiation Revisited
- Using the Chain Rule to find $\frac{dy}{dx}$ for $F(x,y) = 0$ .

Visual Anchors

Dependency Tree for One Independent Variable

This diagram shows how $t$ influences $z$ through two different intermediate paths.

Loading Diagram...

Figure 1 — Mermaid diagram

Dependency Tree for Two Independent Variables

When we have two independent variables ( $u, v$ ), we must follow paths to the specific variable of interest.

Loading Diagram...

Figure 2 — Mermaid diagram

Definition-Example Pairs

1. Total Derivative via Chain Rule

Definition: The derivative of a function with respect to an independent variable when all intermediate variables are also functions of that independent variable.
Example: If $z = x^2 + y^2$ (a paraboloid) and a particle moves along a path $x = \sin(t), y = \cos(t)$ , the total derivative $\frac{dz}{dt}$ tells us how the height of the particle changes over time.

2. Generalized Chain Rule Summation

Definition: For a function $w = f(x, y, z)$ , the rate of change with respect to an input $u$ is the sum of the partials of $w$ with respect to each coordinate, each multiplied by how that coordinate changes with $u$ .
Example: In a weather model where temperature $T$ depends on latitude, longitude, and altitude, and each of those depends on the time of day, the Chain Rule calculates the temperature change rate for a moving weather balloon.

Worked Examples

Example 1: One Independent Variable

Problem: Calculate $\frac{dz}{dt}$ given $z = x^2y - y^2$ , where $x = ext{sin}(t)$ and $y = e^t$ .

Step 1: Find necessary partials.

$\frac{\partial z}{\partial x} = 2xy$
$\frac{\partial z}{\partial y} = x^2 - 2y$
$\frac{dx}{dt} = \cos(t)$
$\frac{dy}{dt} = e^t$

Step 2: Apply the Chain Rule formula. $\frac{dz}{dt} = (2xy)(\cos t) + (x^2 - 2y)(e^t)$

Step 3: Substitute $x$ and $y$ back to get a function of $t$ . $\frac{dz}{dt} = 2\sin(t)e^t\cos(t) + (\sin^2(t) - 2e^t)e^t$

Example 2: Two Independent Variables

Problem: Calculate $\frac{\partial z}{\partial u}$ for $z = x^2 + 3xy$ , where $x = u \cdot v$ and $y = u + v$ .

Step 1: Identify components.

$\frac{\partial z}{\partial x} = 2x + 3y$
$\frac{\partial z}{\partial y} = 3x$
$\frac{\partial x}{\partial u} = v$
$\frac{\partial y}{\partial u} = 1$

Step 2: Calculate. $\frac{\partial z}{\partial u} = (2x + 3y)(v) + (3x)(1)$ $\frac{\partial z}{\partial u} = (2uv + 3(u+v))v + 3uv = 2uv^2 + 3uv + 3v^2 + 3uv = 2uv^2 + 6uv + 3v^2$

Checkpoint Questions

Identify: In the dependency $w = f(x, y, z)$ where $x=g(r, s)$ , $y=h(r, s)$ , and $z=k(r, s)$ , which variables are independent and which are intermediate?
Logic: Why do we add the terms in the Chain Rule formula rather than multiplying the entire expression?
Visualization: Draw a tree diagram for $w = f(x, y)$ where $x = g(u, v)$ and $y = h(u, v)$ . How many paths lead to $v$ ?
Application: If $z$ is a function of $x$ and $y$ , and $x$ and $y$ are functions of $t$ , what is the difference between $\frac{\partial z}{\partial x}$ and $\frac{dz}{dt}$ ?

▶Click to view answers

Independent: $r, s$ . Intermediate: $x, y, z$ .
Because each intermediate variable provides an additive contribution to the total change of the output.
Two paths lead to $v$ (one through $x$ , one through $y$ ).
$\frac{\partial z}{\partial x}$ is the rate of change of $z$ as only $x$ changes; $\frac{dz}{dt}$ is the total rate of change of $z$ as $t$ flows through both $x$ and $y$ .

The Multivariable Chain Rule: A Comprehensive Study Guide

Learning Objectives

After studying this chapter, you should be able to:

State the chain rules for one or two independent variables.
Construct and interpret tree diagrams to visualize dependency paths.
Differentiate between intermediate and independent variables in a composite function.
Perform implicit differentiation of functions involving several variables.
Apply the generalized chain rule to functions with $n$ independent variables and $m$ intermediate variables.

Key Terms & Glossary

Independent Variable: The fundamental inputs at the "bottom" of the dependency chain (e.g., $t$ in $x(t)$ or $u, v$ in $x(u,v)$ ).
Intermediate Variable: A variable that depends on others but is itself an input for a higher-level function (e.g., $x$ and $y$ in $z = f(x,y)$ where $x$ and $y$ are functions of $t$ ).
Composite Function: A function formed by substituting one function into another.
Partial Derivative: The derivative of a multivariable function with respect to one variable while holding others constant.
Total Derivative: The rate of change of a function with respect to an independent variable, accounting for all paths through intermediate variables.

The "Big Idea"

Formula / Concept Box

Case	Dependency	Formula
One Independent Variable	$z = f(x,y)$ , where $x=g(t), y=h(t)$	$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$
Two Independent Variables	$z = f(x,y)$ , where $x=g(u,v), y=h(u,v)$	$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$
Generalized Case	$z = f(x_1, ..., x_n)$ , $x_i = g_i(t_1, ..., t_m)$	$\frac{\partial z}{\partial t_j} = \sum_{i=1}^{n} \frac{\partial z}{\partial x_i} \frac{\partial x_i}{\partial t_j}$

[!IMPORTANT] Note the notation difference: If the final result depends on only one variable, we use $d$ (total derivative). If it depends on multiple variables, we use $\partial$ (partial derivative).

Hierarchical Outline

I. Chain Rule for One Independent Variable
- Dependency Path: $t \to (x, y) \to z$
- Summation Principle: Multiply derivatives along paths; add the paths together.
II. Chain Rule for Two Independent Variables
- Independence: $u$ and $v$ are independent; changing $u$ does not change $v$ .
- Partial Differentiation: We calculate $\frac{\partial z}{\partial u}$ and $\frac{\partial z}{\partial v}$ separately.
III. The Generalized Chain Rule
- Dimensionality: Works for $n$ variables and $m$ parameters.
- Matrix View: Can be represented as a product of Jacobian matrices.
IV. Implicit Differentiation Revisited
- Using the Chain Rule to find $\frac{dy}{dx}$ for $F(x,y) = 0$ .

Visual Anchors

Dependency Tree for One Independent Variable

This diagram shows how $t$ influences $z$ through two different intermediate paths.

Loading Diagram...

Figure 1 — Mermaid diagram

Dependency Tree for Two Independent Variables

When we have two independent variables ( $u, v$ ), we must follow paths to the specific variable of interest.

Loading Diagram...

Figure 2 — Mermaid diagram

Definition-Example Pairs

1. Total Derivative via Chain Rule

Definition: The derivative of a function with respect to an independent variable when all intermediate variables are also functions of that independent variable.
Example: If $z = x^2 + y^2$ (a paraboloid) and a particle moves along a path $x = \sin(t), y = \cos(t)$ , the total derivative $\frac{dz}{dt}$ tells us how the height of the particle changes over time.

2. Generalized Chain Rule Summation

Definition: For a function $w = f(x, y, z)$ , the rate of change with respect to an input $u$ is the sum of the partials of $w$ with respect to each coordinate, each multiplied by how that coordinate changes with $u$ .
Example: In a weather model where temperature $T$ depends on latitude, longitude, and altitude, and each of those depends on the time of day, the Chain Rule calculates the temperature change rate for a moving weather balloon.

Worked Examples

Example 1: One Independent Variable

Problem: Calculate $\frac{dz}{dt}$ given $z = x^2y - y^2$ , where $x = ext{sin}(t)$ and $y = e^t$ .

Step 1: Find necessary partials.

$\frac{\partial z}{\partial x} = 2xy$
$\frac{\partial z}{\partial y} = x^2 - 2y$
$\frac{dx}{dt} = \cos(t)$
$\frac{dy}{dt} = e^t$

Step 2: Apply the Chain Rule formula. $\frac{dz}{dt} = (2xy)(\cos t) + (x^2 - 2y)(e^t)$

Step 3: Substitute $x$ and $y$ back to get a function of $t$ . $\frac{dz}{dt} = 2\sin(t)e^t\cos(t) + (\sin^2(t) - 2e^t)e^t$

Example 2: Two Independent Variables

Problem: Calculate $\frac{\partial z}{\partial u}$ for $z = x^2 + 3xy$ , where $x = u \cdot v$ and $y = u + v$ .

Step 1: Identify components.

$\frac{\partial z}{\partial x} = 2x + 3y$
$\frac{\partial z}{\partial y} = 3x$
$\frac{\partial x}{\partial u} = v$
$\frac{\partial y}{\partial u} = 1$

Step 2: Calculate. $\frac{\partial z}{\partial u} = (2x + 3y)(v) + (3x)(1)$ $\frac{\partial z}{\partial u} = (2uv + 3(u+v))v + 3uv = 2uv^2 + 3uv + 3v^2 + 3uv = 2uv^2 + 6uv + 3v^2$

Checkpoint Questions

Identify: In the dependency $w = f(x, y, z)$ where $x=g(r, s)$ , $y=h(r, s)$ , and $z=k(r, s)$ , which variables are independent and which are intermediate?
Logic: Why do we add the terms in the Chain Rule formula rather than multiplying the entire expression?
Visualization: Draw a tree diagram for $w = f(x, y)$ where $x = g(u, v)$ and $y = h(u, v)$ . How many paths lead to $v$ ?
Application: If $z$ is a function of $x$ and $y$ , and $x$ and $y$ are functions of $t$ , what is the difference between $\frac{\partial z}{\partial x}$ and $\frac{dz}{dt}$ ?

▶Click to view answers

Independent: $r, s$ . Intermediate: $x, y, z$ .
Because each intermediate variable provides an additive contribution to the total change of the output.
Two paths lead to $v$ (one through $x$ , one through $y$ ).
$\frac{\partial z}{\partial x}$ is the rate of change of $z$ as only $x$ changes; $\frac{dz}{dt}$ is the total rate of change of $z$ as $t$ flows through both $x$ and $y$ .