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Chapter 14 – Partial Derivatives14.5 The Chain Rule

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

dy dy dx

dt dx dt

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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

dz z dx z dy

dt x dt y dt

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14.5 The Chain Rule 5

Example 1 – pg. 930 # 2Use the chain rule to find dz/dt or dw/dt.

tytxyxz /1,5,4cos 4

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14.5 The Chain Rule 6

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

2 2 2ln , sin , cos , tanw x y z x t y t z t

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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 DiagramThen, 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 DiagramTo find ∂z/∂s, we find the product of the partial

derivatives along each path from z to s and then add these products:

z z x z y

s x s y s

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14.5 The Chain Rule 11

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

tsvtsuvuz 23,32),/tan(

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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 4Use the Chain Rule to find the indicated

partial derivatives.

1 when ,,

;2,2,2,ln 222

yxy

R

x

R

xywyxvyxuwvuR

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14.5 The Chain Rule 14

Example 5Use the Chain Rule to find the indicated

partial derivatives.

0,1 when ,,

;,,,tan 1

strt

Y

s

Y

r

Y

rtwtsvsruuvwY

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14.5 The Chain Rule 15

Implicit DifferentiationThe 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:

0F dx F dy

x dx y dx

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14.5 The Chain Rule 17

Implicit DifferentiationHowever, dx/dx = 1.So, if ∂F/∂y ≠ 0, we solve for dy/dx

and obtain:

x

y

FFdy x

Fdx Fy

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14.5 The Chain Rule 18

Implicit DifferentiationNow, 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:

0F x F y F z

x x y x z x

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14.5 The Chain Rule 19

Implicit DifferentiationHowever,

So, that equation becomes:

( ) 1 and ( ) 0x yx x

0F F z

x z x

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14.5 The Chain Rule 20

Implicit DifferentiationIf ∂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.

FFz z yx

F Fx yz z

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14.5 The Chain Rule 21

Example 6Use equation 7 to find ∂z/∂x and ∂z/∂y .

zxyz ln

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14.5 The Chain Rule 22

More Examples

The video examples below are from section 14.5 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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