1. Chapter 14 – Partial Derivatives
14.5 The Chain Rule
Objectives:
How to use the Chain Rule and applying it to applications
How to use the Chain Rule for Implicit Differentiation
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14.5 The Chain Rule

2. Chain Rule: Single Variable Functions
Recall that the Chain Rule for functions of a single variable gives the following rule for differentiating a composite function.
If y = f (x) and x = g (t), where f and g are differentiable functions, then y is indirectly a differentiable function of t, and
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14.5 The Chain Rule

3. Chain Rule: Multivariable Functions
For functions of more than one variable, the Chain Rule has several versions.
Each gives a rule for differentiating a composite function.
The first version (Theorem 2) deals with the case where z = f (x, y) and each of the variables x and y is, in turn, a function of a variable t.
This means that z is indirectly a function of t, z = f (g(t), h(t)), and the Chain Rule gives a formula for differentiating z as a function of t.
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14.5 The Chain Rule

4. Chain Rule: Case 1
Since we often write ∂z/∂x in place of ∂f/∂x, we can rewrite the Chain Rule in the form
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14.5 The Chain Rule

5. Example 1 – pg. 930 # 2 Use the chain rule to find dz/dt or dw/dt.
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14.5 The Chain Rule

6. Example 2 – pg. 930 # 6 Use the chain rule to find dz/dt or dw/dt.
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14.5 The Chain Rule

7. Chain Rule: Case 2
Case 2 of the Chain Rule contains three types of variables:
s and t are independent variables.
x and y are called intermediate variables.
z is the dependent variable.
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14.5 The Chain Rule

8. Using a Tree Diagram with Chain Rule
We draw branches from the dependent variable z to the intermediate variables x and y to indicate that z is a function of x and y.
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14.5 The Chain Rule

9. Tree Diagram
Then, we draw branches from x and y to the independent variables s and t.
On each branch, we write the corresponding partial derivative.
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14.5 The Chain Rule

10. Tree Diagram
To find ∂z/∂s, we find the product of the partial derivatives along each path from z to s and then add these products:
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14.5 The Chain Rule

11. Example 3 – pg. 930 # 12 Use the Chain rule to find ∂z/∂s and ∂z/∂t.
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14.5 The Chain Rule

12. Chain Rule: General Version
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14.5 The Chain Rule

13. Example 4
Use the Chain Rule to find the indicated partial derivatives.
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14.5 The Chain Rule

14. Example 5
Use the Chain Rule to find the indicated partial derivatives.
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14.5 The Chain Rule

15. Implicit Differentiation
The Chain Rule can be used to give a more complete description of the process of implicit differentiation that was introduced in Sections 3.5 and 14.3
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14.5 The Chain Rule

16. Implicit Differentiation
If F is differentiable, we can apply Case 1 of the Chain Rule to differentiate both sides of the equation F(x, y) = 0 with respect to x.
Since both x and y are functions of x, we obtain:
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14.5 The Chain Rule

17. Implicit Differentiation
However, dx/dx = 1.
So, if ∂F/∂y ≠ 0, we solve for dy/dx and obtain:
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14.5 The Chain Rule

18. Implicit Differentiation
Now, we suppose that z is given implicitly as a function z = f(x, y) by an equation of the form F(x, y, z) = 0.
This means that F(x, y, f(x, y)) = 0 for all (x, y) in the domain of f.
If F and f are differentiable, then we can use the Chain Rule to differentiate the equation F(x, y, z) = 0 as follows:
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14.5 The Chain Rule

19. Implicit Differentiation
However,
So, that equation becomes:
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14.5 The Chain Rule

20. Implicit Differentiation
If ∂F/∂z ≠ 0, we solve for ∂z/∂x and obtain the first formula in these equations.
The formula for ∂z/∂y is obtained in a similar manner.
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14.5 The Chain Rule

21. Example 6 Use equation 7 to find ∂z/∂x and ∂z/∂y . Dr. Erickson
14.5 The Chain Rule

22. More Examples
The video examples below are from section in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 2
Example 4
Example 5
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14.5 The Chain Rule