Section 13.6
The Chain Rule
Navigating the "Tree of Variables" to find derivatives.
1
Introduction
In 1D, if and , then .
In 3D, if and , changes in affect through both and .
Total Change = (Change via x) + (Change via y)
2
Case 1: One Independent Variable
Tree Diagram
z
∂z/∂x
∂z/∂y
x
y
dx/dt
dy/dt
t
Formula:
3
Case 2: Two Independent Variables
If and , then depends on both and .
Partial wrt s:
Partial wrt t:
4
Worked Examples
Example 1: Single Independent Variable
Let , where and . Find .
1. Partials: .
2. Derivatives of paths: .
3. Apply Formula:
4. Substitute back (optional unless requested):
.
Check: . . It works!
Example 2: Two Independent Variables
Let , , . Find .
1. , .
2. , .
3. Formula:
.
Example 3: Implicit Differentiation
Find for .
Let .
The formula is .
1. .
2. .
.
Answer: .
5
Practice Quiz
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