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. Section 2.5
The Chain Rule
V63.0121.041, Calculus I
New York University
October 6, 2010
Announcements
Quiz 2 in recitation next week (October 11-15)
No class Monday, October 11
Midterm in class Monday, October 18 on §§1.1–2.5
. . . . . .
2. Announcements
Quiz 2 in recitation next
week (October 11-15)
No class Monday, October
11
Midterm in class Monday,
October 18 on §§1.1–2.5
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 2 / 36
3. Objectives
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 3 / 36
4. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
5. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
x
. g
. (x)
g
. .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
6. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
x
. g
. (x)
g
. . f
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
7. Compositions
See Section 1.2 for review
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
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
8. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
g
. (x) f
.(g(x))
. ◦ g
x
.
g
. f . f
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
9. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
g
. (x) f
.(g(x))
. ◦ g
x
.
g
. f . 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 4 / 36
10. Outline
Heuristics
Analogy
The Linear Case
The chain rule
Examples
Related rates of change
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 5 / 36
11. Analogy
Think about riding a bike. To go
faster you can either:
.
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
12. Analogy
Think about riding a bike. To go
faster you can either:
pedal faster
.
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
13. Analogy
Think about riding a bike. To go
faster you can either:
pedal faster
change gears
.
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
14. Analogy
Think about riding a bike. To go
faster you can either:
pedal faster
change gears
.
The angular position (φ) of the back wheel depends on the position of
the front sprocket (θ):
R.θ
.
φ(θ) =
r.
.
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
15. Analogy
Think about riding a bike. To go
faster you can either:
pedal faster
change gears
r
. adius of front sprocket .
The angular position (φ) of the back wheel depends on the position of
the front sprocket (θ):
R.θ
.
φ(θ) =
r.
.
r
. adius of back sprocket
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
16. Analogy
Think about riding a bike. To go
faster you can either:
pedal faster
change gears
.
The angular position (φ) of the back wheel depends on the position of
the front sprocket (θ):
R.θ
.
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.
. . . . . .
.
Image credit: SpringSun
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 6 / 36
17. The Linear Case
Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 7 / 36
18. The Linear Case
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)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 7 / 36
19. The Linear Case
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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 7 / 36
20. The Linear Case
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 7 / 36
21. The Linear Case
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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 7 / 36
22. The Nonlinear Case
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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 8 / 36
23. Outline
Heuristics
Analogy
The Linear Case
The chain rule
Examples
Related rates of change
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 9 / 36
24. Theorem of the day: The chain rule
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(x). Then
dy dy du
=
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 10 / 36
25. Observations
Succinctly, the derivative of a
composition is the product of
the derivatives
.
. . . . . .
.
Image credit: ooOJasonOoo
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 11 / 36
26. Theorem of the day: The chain rule
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(x). Then
dy dy du
=
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 12 / 36
27. Observations
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
.
. . . . . .
.
Image credit: ooOJasonOoo
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 13 / 36
28. Compositions
See Section 1.2 for review
Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do
g first, then f.”
g
. (x) f
.(g(x))
. ◦ g
x
.
g
. f . f
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 14 / 36
29. Observations
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
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 15 / 36
30. Theorem of the day: The chain rule
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(x). Then
dy dy du
=
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 16 / 36
31. Theorem of the day: The chain rule
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)
dy
In Leibnizian notation, let y = f(u) and u = g(x).du
Then
. .
dx
du
dy dy du
=
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 16 / 36
32. Outline
Heuristics
Analogy
The Linear Case
The chain rule
Examples
Related rates of change
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 17 / 36
33. Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 18 / 36
34. Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
Solution
First, write h as f ◦ g.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 18 / 36
35. Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
Solution
√
First, write h as f ◦ g. Let f(u) = u and g(x) = 3x2 + 1.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 18 / 36
36. Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
Solution
√
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
h′ (x) = 1 u−1/2 (6x)
2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 18 / 36
37. Example
Example
√
let h(x) = 3x2 + 1. Find h′ (x).
Solution
√
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
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 18 / 36
38. Corollary
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 ) = nun−1 .
dx dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 19 / 36
39. Does order matter?
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 20 / 36
40. Does order matter?
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx
Solution
For the first, let u = 4x and y = sin(u). Then
dy dy du
= · = cos(u) · 4 = 4 cos 4x.
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 20 / 36
41. Does order matter?
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx
Solution
For the first, let u = 4x and y = sin(u). Then
dy dy du
= · = cos(u) · 4 = 4 cos 4x.
dx du dx
For the second, let u = sin x and y = 4u. Then
dy dy du
= · = 4 · cos x
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 20 / 36
42. Order matters!
Example
d d
Find (sin 4x) and compare it to (4 sin x).
dx dx
Solution
For the first, let u = 4x and y = sin(u). Then
dy dy du
= · = cos(u) · 4 = 4 cos 4x.
dx du dx
For the second, let u = sin x and y = 4u. Then
dy dy du
= · = 4 · cos x
dx du dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 20 / 36
43. Example
(√ )2
. Find f′ (x).
3
Let f(x) = x5 − 2 + 8
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 21 / 36
44. Example
(√ )2
. Find f′ (x).
3
Let f(x) = x5 − 2 + 8
Solution
d (√ 5
3
)2 (√
3
) d (√
3
)
x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8
dx dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 21 / 36
49. A metaphor
Think about peeling an onion:
(√ )2
3
f(x) = x 5
−2 +8
5
√
3
+8
.
(√ )
2
f′ (x) = 2 x5 − 2 + 8 1 (x5 − 2)−2/3 (5x4 )
3
3
. . . . . .
.
Image credit: photobunny
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 22 / 36
50. Combining techniques
Example
d ( 3 )
Find (x + 1)10 sin(4x2 − 7)
dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 23 / 36
51. Combining techniques
Example
d ( 3 )
Find (x + 1)10 sin(4x2 − 7)
dx
Solution
The “last” part of the function is the product, so we apply the product
rule. Each factor’s derivative requires the chain rule:
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 23 / 36
52. Combining techniques
Example
d ( 3 )
Find (x + 1)10 sin(4x2 − 7)
dx
Solution
The “last” part of the function is the product, so we apply the product
rule. Each factor’s derivative requires the chain rule:
d ( 3 )
(x + 1)10 · sin(4x2 − 7)
dx ( ) ( )
d 3 d
= (x + 1) 10
· sin(4x − 7) + (x + 1) ·
2 3 10
sin(4x − 7)
2
dx dx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 23 / 36
53. Combining techniques
Example
d ( 3 )
Find (x + 1)10 sin(4x2 − 7)
dx
Solution
The “last” part of the function is the product, so we apply the product
rule. Each factor’s derivative requires the chain rule:
d ( 3 )
(x + 1)10 · sin(4x2 − 7)
dx ( ) ( )
d 3 d
= (x + 1) 10
· sin(4x − 7) + (x + 1) ·
2 3 10
sin(4x − 7)
2
dx dx
= 10(x3 + 1)9 (3x2 ) sin(4x2 − 7) + (x3 + 1)10 · cos(4x2 − 7)(8x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 23 / 36
54. Your Turn
Find derivatives of these functions:
1. y = (1 − x2 )10
√
2. y = sin x
√
3. y = sin x
4. y = (2x − 5)4 (8x2 − 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))))))
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 24 / 36
55. Solution to #1
Example
Find the derivative of y = (1 − x2 )10 .
Solution
y′ = 10(1 − x2 )9 (−2x) = −20x(1 − x2 )9
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 25 / 36
56. Solution to #2
Example
√
Find the derivative of y = sin x.
Solution
√
Writing sin x as (sin x)1/2 , we have
cos x
y′ = 1
2 (sin x)−1/2 (cos x) = √
2 sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 26 / 36
57. Solution to #3
Example
√
Find the derivative of y = sin x.
Solution
(√ )
′ d 1/2 1 −1/2 cos x
y = 1/2
sin(x ) = cos(x ) 2 x = √
dx 2 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 27 / 36
58. Solution to #4
Example
Find the derivative of y = (2x − 5)4 (8x2 − 5)−3
Solution
We need to use the product rule and the chain rule:
y′ = 4(2x − 5)3 (2)(8x2 − 5)−3 + (2x − 5)4 (−3)(8x2 − 5)−4 (16x)
The rest is a bit of algebra, useful if you wanted to solve the equation
y′ = 0:
[ ]
y′ = 8(2x − 5)3 (8x2 − 5)−4 (8x2 − 5) − 6x(2x − 5)
( )
= 8(2x − 5)3 (8x2 − 5)−4 −4x2 + 30x − 5
( )
= −8(2x − 5)3 (8x2 − 5)−4 4x2 − 30x + 5
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 28 / 36
59. Solution to #5
Example
√
z−1
Find the derivative of F(z) = .
z+1
Solution
( )−1/2 ( )
1 z−1(z + 1)(1) − (z − 1)(1)
y′ =
2 z+1 (z + 1)2
( )1/2 ( )
1 z+1 2 1
= =
2 z−1 (z + 1)2 (z + 1)3/2 (z − 1)1/2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 29 / 36
60. Solution to #6
Example
Find the derivative of y = tan(cos x).
Solution
y′ = sec2 (cos x) · (− sin x) = − sec2 (cos x) sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 30 / 36
61. Solution to #7
Example
Find the derivative of y = csc2 (sin θ).
Solution
Remember the notation:
y = csc2 (sin θ) = [csc(sin θ)]2
So
y′ = 2 csc(sin θ) · [− csc(sin θ) cot(sin θ)] · cos(θ)
= −2 csc2 (sin θ) cot(sin θ) cos θ
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 31 / 36
62. Solution to #8
Example
Find the derivative of y = sin(sin(sin(sin(sin(sin(x)))))).
Solution
Relax! It’s just a bunch of chain rules. All of these lines are multiplied
together.
y′ = cos(sin(sin(sin(sin(sin(x))))))
· cos(sin(sin(sin(sin(x)))))
· cos(sin(sin(sin(x))))
· cos(sin(sin(x)))
· cos(sin(x))
· cos(x))
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 32 / 36
63. Outline
Heuristics
Analogy
The Linear Case
The chain rule
Examples
Related rates of change
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 33 / 36
64. Related rates of change at the Deli
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
dV dV
per minute. Then for the machine is 5 times greater than for
dt dt
the deli clerk. This is explained by the
A. chain rule
B. product rule
C. quotient Rule
D. addition rule
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 34 / 36
65. Related rates of change at the Deli
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
dV dV
per minute. Then for the machine is 5 times greater than for
dt dt
the deli clerk. This is explained by the
A. chain rule
B. product rule
C. quotient Rule
D. addition rule
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 34 / 36
66. Related rates of change in the ocean
Question
The area of a circle, A = πr2 ,
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
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 35 / 36
67. Related rates of change in the ocean
Question
The area of a circle, A = πr2 ,
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
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 35 / 36
68. Summary
The derivative of a
composition is the product
of derivatives
In symbols:
(f ◦ g)′ (x) = f′ (g(x))g′ (x)
Calculus is like an onion,
and not because it makes
you cry!
. . . . . .
V63.0121.041, Calculus I (NYU) Section 2.5 The Chain Rule October 6, 2010 36 / 36