The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

1. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Notes
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Sec on 2.5
The Chain Rule
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
February 23, 2011
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Notes
Announcements
Quiz 2 next week on
§§1.5, 1.6, 2.1, 2.2
Midterm March 7 on all
sec ons in class (covers
all sec ons up to 2.5)
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Notes
Objectives
Given a compound
expression, write it as a
composi on of func ons.
Understand and apply
the Chain Rule for the
deriva ve of a
composi on of func ons.
Understand and use
Newtonian and Leibnizian
nota ons for the Chain
Rule.
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2. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Compositions Notes
See Section 1.2 for review
Defini on
If f and g are func ons, the composi on (f ◦ g)(x) = f(g(x)) means “do g first,
then f.”
x
g
f◦g
g(x)
. f
f(g(x))
Our goal for the day is to understand how the deriva ve of the composi on of
two func ons depends on the deriva ves of the individual func ons.
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Notes
Outline
Heuris cs
Analogy
The Linear Case
The chain rule
Examples
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Notes
Analogy
Think about riding a bike. To
go faster you can either:
pedal faster
change gears
radius of front sprocket .
The angular posi on (φ) of the back wheel depends on the posi on
of the front sprocket (θ):
R..
θ
φ(θ) =
r..
And so the angular speed of the back wheel depends on the
radius of back sprocket
deriva ve of this func on and the speed of the front sprocket.
. Image credit: SpringSun
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3. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Notes
The Linear Case
Ques on
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composi on?
Answer
f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b)
The composi on is also linear
The slope of the composi on is the product of the slopes of the two
func ons.
The deriva ve is supposed to be a local lineariza on of a func on. So there
should be an analog of this property in deriva ves.
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Notes
The Nonlinear Case
Let u = g(x) and y = f(u). Suppose x is changed by a small amount
∆x. Then
∆y
f′ (y) ≈ =⇒ ∆y ≈ f′ (y)∆u
∆x
and
∆u
g′ (y) ≈ =⇒ ∆u ≈ g′ (u)∆x.
∆x
So
∆y
∆y ≈ f′ (y)g′ (u)∆x =⇒ ≈ f′ (y)g′ (u)
∆x
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Notes
Outline
Heuris cs
Analogy
The Linear Case
The chain rule
Examples
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4. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Notes
Theorem of the day: The chain rule
Theorem
Let f and g be func ons, with g differen able at x and f differen able
at g(x). Then f ◦ g is differen able at x and
(f ◦ g)′ (x) = f′ (g(x))g′ (x)
In Leibnizian nota on, let y = f(u) and u = g(x). Then
dy dy du
=
dx du dx
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Notes
Observations
Succinctly, the deriva ve of a
composi on is the product
of the deriva ves
The only complica on is
where these deriva ves are
evaluated: at the same point
the func ons are
In Leibniz nota on, the Chain
Rule looks like cancella on of .
(fake) frac ons
. Image credit: ooOJasonOoo
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Notes
Outline
Heuris cs
Analogy
The Linear Case
The chain rule
Examples
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5. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Notes
Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
Solu on
√
First, write h as f ◦ g. Let f(u) = u and g(x) = 3x2 + 1. Then
f′ (u) = 1 u−1/2 , and g′ (x) = 6x. So
2
3x
h′ (x) = 1 u−1/2 (6x) = 1 (3x2 + 1)−1/2 (6x) = √
2 2
3x2 + 1
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Notes
Corollary
Corollary (The Power Rule Combined with the Chain Rule)
If n is any real number and u = g(x) is differen able, then
d n du
(u ) = nun−1 .
dx dx
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Notes
Does order matter?
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx
Solu on
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6. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Example Notes
(√ )2
x − 2 + 8 . Find f′ (x).
3 5
Let f(x) =
Solu on
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Notes
A metaphor
Think about peeling an onion:
(√ )2
3
f(x) = x5 −2 +8
5
√
3
+8 .
2
(√ )
f′ (x) = 2 x5 − 2 + 8 1 5
− 2)−2/3 (5x4 )
3
3 (x
. Image credit: photobunny
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Notes
Combining techniques
Example
d ( 3 )
Find (x + 1)10 sin(4x2 − 7)
dx
Solu on
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7. . V63.0121.001: Calculus I
. Sec on 2.5: The Chain Rule
. February 23, 2011
Notes
Related rates of change in the ocean
Ques on
The area of a circle, A = πr2 , changes as its radius
changes. If the radius changes with respect to me,
the change in area with respect to me is
dA
A. = 2πr
dr
dA dr
B. = 2πr +
dt dt
dA dr
C. = 2πr
dt dt
.
D. not enough informa on
.
Image credit: Jim Frazier
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Notes
Summary
The deriva ve of a
composi on is the
product of deriva ves
In symbols:
(f ◦ g)′ (x) = f′ (g(x))g′ (x)
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Notes
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