MAT 1236Calculus III

Section 14.5

The Chain Rule

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HW..

WebAssign 14.5 Quiz: 14.5

Preview

The Chain Rule for multivariable functions

Look only at how to memorize the formulas

Why the formulas actually work depends on 14.4 which we do not cover.

The Doppler Effect

sf

0,o sf v t v t

sv t

0v t

1 variable Vs 2 variables

)(xfy ),( yxfz

The Chain Rule

dx

du

du

dy

dx

dy

xgfy

xguufy

))((Therefore,

)( ),(

dy

dudy

dxd

y

u

x

u

dx

The Chain Rule: Case 1

, , , z f x y x g t y h t

dz

dt

z

t

y

z

x

z

y

dx

dt

x

dy

dt

Example 1

Find

2 2ln , sin , tz x y x t y e

dz

dt

z

t

y

z

x

z

y

dx

dt

x

dy

dt

The Chain Rule: Case 2 , , , , ,z f x y x g s t y h s t

z

s

z

t

z

t

yx

tss

z

x

z

y

x

s

x

t

y

s

y

t

Example 2

Find

2 2ln , sin , stz x y x s t y e

z

s

z

t

yx

tss

z

x

z

y

x

s

x

t

y

s

y

t

Other Cases

Similar

Example 3

Find in terms of partial derivatives in x and y (i.e . )

Note that the function f is not given explicitly.

2 2, , , u f x y x s t y s t 2

2, u u

s s

2 2 2

2 2, , , , u u u u u

x y x x y y

Example 4 (Implicit Differentiation)

Suppose y is a function in x and is given implicitly by the equation

What is the relationship between

, 0F x y

, , and ?x y

dyF F

dx

Example 4 (Implicit Differentiation)

Suppose y is a function in x and is given implicitly by the equation

What is the relationship between

3 2 2 0x y xy

, , and ?x y

dyF F

dx

Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the

equation

3 2 2 0x y xy

Example 4 (Implicit Differentiation)

Suppose y is a function in x and is given implicitly by the equation

Show that

, 0F x y

x

y

Fdy

dx F

Where to use? (If time permitted)

Taken from MAT 3724 Applied Analysis (Mathematical Physics)

Wave Equation: Set Up

x

displacement ( , )u x t

Assumptions:1.2.3.

Thought Experiment

x

displacement ( , )u x t

2 0 , 0 (2.12)

( ,0) ( ), ( ,0) ( ) (2.13)tt xx

t

u c u x R t

u x f x u x g x x R

Thought Experiment

What would you expect to happen?

x0

Thought Experiment

Thought Experiment

Modeling Wave Propagation in a String

Modeling Wave Propagation in a String

Solution: d’Alembert’s Formula

x

displacement ( , )u x t

2 0 , 0 (2.12)

( ,0) ( ), ( ,0) ( ) (2.13)

1 1( , ) ( ) ( ) ( )

2 2

tt xx

t

x ct

x ct

u c u x R t

u x f x u x g x x R

u x t f x ct f x ct g s dsc