Lesson 10: The Chain Rule (Section 21 handout)

V63.0121.021, Calculus I Section 2.5 : The Chain Rule October 7, 2010

Notes
Section 2.5
The Chain Rule

V63.0121.021, Calculus I

New York University

October 7, 2010

Announcements

Quiz 2 in recitation next week (October 11-15)
Midterm in class Tuesday, october 19 on §§1.1–2.5

Announcements
Notes

Quiz 2 in recitation next
week (October 11-15)
Midterm in class Tuesday,
october 19 on §§1.1–2.5

V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 2 / 36

Objectives
Notes

Given a compound
expression, write it as a
composition of functions.
Understand and apply the
Chain Rule for the derivative
of a composition of
functions.
Understand and use
Newtonian and Leibnizian
notations for the Chain Rule.


1


Compositions
See Section 1.2 for review Notes

Deﬁnition
If f and g are functions, the composition (f ◦ g )(x) = f (g (x)) means “do
g ﬁrst, then f .”

x g (x) f (g (x))
g f ◦ g f

Our goal for the day is to understand how the derivative of the composition
of two functions depends on the derivatives of the individual functions.


Outline
Notes

Heuristics
Analogy
The Linear Case

The chain rule

Examples

Related rates of change


Analogy
Notes

Think about riding a bike. To go
faster you can either:
pedal faster
change gears

radius of front sprocket
The angular position (ϕ) of the back wheel depends on the position of the
front sprocket (θ):
Rθ
ϕ(θ) =
r
And so the angular speed of the back wheel depends on the derivative of
this function and the speed of the front sprocket.
radius of back sprocket
Image credit: SpringSun

2


The Linear Case
Notes

Question
Let f (x) = mx + b and g (x) = m x + b . What can you say about the
composition?

Answer

f (g (x)) = m(m x + b ) + b = (mm )x + (mb + b)
The composition is also linear
The slope of the composition is the product of the slopes of the two
functions.

The derivative is supposed to be a local linearization of a function. So
there should be an analog of this property in derivatives.


The Nonlinear Case
Notes

Let u = g (x) and y = f (u). Suppose x is changed by a small amount ∆x.
Then
∆y ≈ f (y )∆u
and
∆u ≈ g (u)∆x.
So
∆y
∆y ≈ f (y )g (u)∆x =⇒ ≈ f (y )g (u)
∆x


Outline
Notes

Heuristics
Analogy
The Linear Case

The chain rule

Examples



3


Theorem of the day: The chain rule
Notes

Theorem
Let f and g be functions, with g differentiable at x and f differentiable at
g (x). Then f ◦ g is differentiable at x and

(f ◦ g ) (x) = f (g (x))g (x)

In Leibnizian notation, let y = f (u) and u = g dy Then
du
(x).
dx
du

dy dy du
=
dx du dx


Observations
Notes

Succinctly, the derivative of a
composition is the product of
the derivatives
The only complication is where
these derivatives are evaluated:
at the same point the functions
are
In Leibniz notation, the Chain
Rule looks like cancellation of
(fake) fractions

Image credit: ooOJasonOoo

Outline
Notes

Heuristics
Analogy
The Linear Case

The chain rule

Examples



4


Example
Notes

Example
let h(x) = 3x 2 + 1. Find h (x).

Solution
√
First, write h as f ◦ g . Let f (u) = u and g (x) = 3x 2 + 1. Then
−1/2
f (u) = 1 u
2 , and g (x) = 6x. So

3x
h (x) = 1 u −1/2 (6x) = 2 (3x 2 + 1)−1/2 (6x) = √
2
1
3x 2 + 1


Corollary
Notes

Corollary (The Power Rule Combined with the Chain Rule)
If n is any real number and u = g (x) is diﬀerentiable, then

d n du
(u ) = nu n−1 .
dx dx


Order matters!
Notes
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx

Solution


5


Example
3
2 Notes
Let f (x) = x5 − 2 + 8 . Find f (x).

Solution


A metaphor
Notes

Think about peeling an onion:
2
3
f (x) = x 5 −2 +8
5

√
3

+8
2

− 2)−2/3 (5x 4 )
3 1 5
f (x) = 2 x5 − 2 + 8 3 (x

Image credit: photobunny

Combining techniques
Notes
Example
d
Find (x 3 + 1)10 sin(4x 2 − 7)
dx

Solution


6


Your Turn
Notes

Find derivatives of these functions:
1. y = (1 − x 2 )10
√
2. y = sin x
√
3. y = sin x
4. y = (2x − 5)4 (8x 2 − 5)−3
z −1
5. F (z) =
z +1
6. y = tan(cos x)
7. y = csc2 (sin θ)
8. y = sin(sin(sin(sin(sin(sin(x))))))


Solution to #1
Notes

Example
Find the derivative of y = (1 − x 2 )10 .

Solution


Solution to #2
Notes

Example
√
Find the derivative of y = sin x.

Solution


7


Solution to #3
Notes

Example
√
Find the derivative of y = sin x.

Solution


Solution to #4
Notes
Example
Find the derivative of y = (2x − 5)4 (8x 2 − 5)−3

Solution


Solution to #5
Notes

Example
z −1
Find the derivative of F (z) = .
z +1

Solution


8


Solution to #6
Notes

Example
Find the derivative of y = tan(cos x).

Solution


Solution to #7
Notes

Example

Solution


Solution to #8
Notes
Example
Find the derivative of y = sin(sin(sin(sin(sin(sin(x)))))).

Solution


9


Outline
Notes

Heuristics
Analogy
The Linear Case

The chain rule

Examples



Related rates of change at the Deli
Notes

Question
Suppose a deli clerk can slice a stick of pepperoni (assume the tapered
ends have been removed) by hand at the rate of 2 inches per minute, while
a machine can slice pepperoni at the rate of 10 inches per minute. Then
dV dV
for the machine is 5 times greater than for the deli clerk. This is
dt dt
explained by the
A. chain rule
B. product rule
C. quotient Rule
D. addition rule


Related rates of change in the ocean
Notes

Question
The area of a circle, A = πr 2 ,
changes as its radius changes. If
the radius changes with respect
to time, the change in area with
respect to time is
dA
A. = 2πr
dr
dA dr
B. = 2πr +
dt dt
dA dr
C. = 2πr
dt dt
D. not enough information

Image credit: Jim Frazier

10


Summary
Notes

The derivative of a
composition is the product
of derivatives
In symbols:
(f ◦ g ) (x) = f (g (x))g (x)


Notes

Notes

11

Lesson 10: The Chain Rule (Section 21 handout)

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