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The Chain (Saw) Rule
Lesson 3.4
The Chain Rule• According to Mrs. Armstrong …
“Pull the chain and the light comes on!”
Introduction
Sludge Falls
• CO2 is changing at rate of 0.02 ppm for each person
• Population growing atrate of 1000 people/yr
• We seek rate of increasing pollution with respect to time
(0.02 ppm/prsn)(1000 people/yr) = 20 ppm/yr
A Composite Function
The level of pollution L is a function of the population P, which is itself a function of time t.
L = f(P(t))
Then L’ … is
Rate of Change of L with
respect to t=
Rate of change of L with
respect to P
Rate of change of P with
respect to t
A Composite FunctionIn Leibniz notation:
dL dL dP
dt dP dt
Result in pollution as a
function of time
Result in pollution as a
function of time Pollution as a function of the
population
Pollution as a function of the
population
Population as a function of
time
Population as a function of
time
The Chain RuleGiven • y = f(u) is a differentiable function of u• u is also a differentiable function … of x• Then y = f(u(x))
• Then
dy dy du
dx du dx
Example• Given:
y = (6x3 – 4x + 7)3
• Then u(x) = 6x3 – 4x + 7and f(u) = u3
• Thusf’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)
dy
du
du
dx
Example• Given g(u) = u5
u(x) = 3x + 1
• Then g’(u) = ?? u’(x) = ??
• f(x) = (3x + 1)5
f’(x) = ??
Example• Find equation of tangent line to
at (2,3)
2( ) 5f x x
Example
• Try
• Which is the u(x), the “inner” function?
• Which is the f(u), the “outer” function?
• What is u’(x), f’(u) ??
2cos(4 7 5)d
x xdx
Example• Try with multiple levels of nested functions
2 3( ) sin ((4 2) )f x x
Example• Try in combination with the product rule
2 3( ) (6 2) xg x x e
Assignment Part A
• Lesson 3.4A
• Page 161
• Exercises 1 – 49 Odd
Derivative for ln x
• Graph the difference function for ln x
This has a familiar look,
pardner!
This has a familiar look,
pardner!
1ln for x 0
dx
dx x
1ln for x 0
dx
dx x
Derivatives for Bases Other Than e
ln
ln
1log
ln
1log
ln
x x
u u
a
a
da a a
dxd dua a a
dx dxd
xdx a x
d duu
dx a u dx
ViewGeogebra
Demo
ViewGeogebra
Demo
ViewGeogebra
Demo
ViewGeogebra
Demo
One More Try
• Determine the derivative
• What is the equation of the tangent line of f(x) when x = 1?
ln( ) '( ) ?
tf t f t
t
( ) lnf x x x
Assignment Part B
• Lesson 3.4B
• Page 162
• Exercises 51 – 127 EOO