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.”
g f
x g(x)
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.”
g f
x g(x) f(g(x))
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.”
g f
x f(g(x))
f◦
g(x
g
)
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.”
g f
x f(g(x))
f◦
g(x
g
)
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.
11. Analogy
Think about riding a bike. To
go faster you can either:
) pedal faster
) change gears
Image credit: SpringSun
12. 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):
And so the angular speed of the back wheel depends on the
derivative of this function and the speed of the front wheel.
Image credit: SpringSun
φ(θ) =
Rθ
r
14. 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)
15. 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
16. 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.
17. 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.
18. The Nonlinear Case
See the Mathematica applet
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 ≈ f′
(y)g′(u)∆x =⇒
∆y
∆x
≈ f′
(y)g′(u)
20. 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
22. 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
23. 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:
24. 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 f
x f(g(x))
f◦
g(x
g
)
25. 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:
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
27. Theorem of the day: The chain rule
dy dy du
dx
=
du dx
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 =
dy
g(
/
d
x
/
u
). Then
/
d
/
u dx
32. Example
Example
let h(x) =
√
3x2 +1. Find h′
(x).
Solution √ 2
First, write h as f◦ g. Let f(u) = u and g(x) = 3x +1. Then
2
f′
(u) = 1u−1/2, and g′(x) = 6x. So
2
h′
(x) = 1 u−1/2(6x)
33. Example
Example
let h(x) =
√
3x2 +1. Find h′
(x).
Solution √ 2
First, write h as f◦ g. Let f(u) = u and g(x) = 3x +1. Then
2
f′
(u) = 1u−1/2, and g′(x) = 6x. So
(x) =
h′
u−
1 1/2
2 2
(6x) = (
1 2 −1/2 3x
3x +1) (6x) = √
3x2 + 1
34. Corollary
Corollary (The Power Rule Combined with the Chain Rule)
If n is any real number and u = g(x) is differentiable, then
d du
dx
(un) = nun−1
dx
.
36. Does order matter?
Example
Find
d
(sin 4x) and compare it to
d
(4 sin x).
dx dx
Solution
) For the first, let u = 4x and y = sin(u). Then
dy dy du
dx
=
du
·
dx
= cos(u)· 4 = 4 cos 4x.
37. Does order matter?
Example
Find
d
(sin 4x) and compare it to
d
(4 sin x).
dx dx
Solution
) For the first, let u = 4x and y = sin(u). Then
dy dy du
dx
=
du
·
dx
= cos(u)· 4 = 4 cos 4x.
) For the second, let u = sin x and y = 4u. Then
dy dy du
dx
=
du
·
dx
= 4 · cos x
38. Order matters!
Example
Find
d
(sin 4x) and compare it to
d
(4 sin x).
dx dx
Solution
) For the first, let u = 4x and y = sin(u). Then
dy dy du
dx
=
du
·
dx
= cos(u)· 4 = 4 cos 4x.
) For the second, let u = sin x and y = 4u. Then
dy dy du
dx
=
du
·
dx
= 4 · cos x
47. Combining techniques
Example
d
dx
(
Find ( 3 10 2
x +1) sin(4x − 7)
)
Solution
The “last” part of the function is the product, so we apply the
product rule. Each factor’s derivative requires the chain rule:
48. Combining techniques
Example
d
dx
(
Find ( 3 10 2
x +1) sin(4x − 7)
)
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
dx
3 10
(x +1) · sin( 2
4x − 7)
( )
(
d
dx
= ( 3 10
)
x +1) ·sin( 2 3 10
4x −7)+(x +1) ·
(
d
dx
sin( 2
4x − 7)
)
49. Combining techniques
Example
d
dx
(
Find ( 3 10 2
x +1) sin(4x − 7)
)
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
dx
3 10
(x +1) · sin( 2
4x − 7)
( )
(
d
dx
= (
)
3 10 2 3 10
x +1) ·sin(4x −7)+(x +1) ·
(
d
dx
2
sin(4x − 7)
)
= 10(x3 +1)9(3x2)sin(4x2 − 7)+(x3 +1)10 · cos(4x2 − 7)(8x)
50. 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
5. F(z) =
√
z − 1
z + 1
6. y = tan(cosx)
7. y = csc2(sin θ)
8. y = sin(sin(sin(sin(sin(sin(x))))))
51. Solution to #1
Example
Find the derivative of y = (1 − x2)10.
Solution
y′ = 10(1 − x2)9(−2x) = −20x(1 − x2)9
52. Solution to #2
Example
Find the derivative of y =
√
sin x.
Solution
√ 1/2
Writing sin x as (sinx) , we have
′ 1 −1/2 cos x
y = 2 (sin x) (cosx) =
2
√
sin x
53. Solution to #3
Example
Find the derivative of y = sin
√
x.
Solution
′ d 1/2 1/2 1 −1/2
y =
dx
sin(x ) = cos(x )2 x =
cos
(√
x
)
2
√
x
54. 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
)
55. Solution to #5
Example
√
Find the derivative of F(z) =
z + 1
z − 1
.
Solution
y′ =
2
(
1 z − 1
z + 1
)−1/2 (
(z + 1)(1) − (z − 1)(1)
(z + 1)2
)
=
2
(
1 z + 1
z − 1
)1/2 (
(z + 1)2 =
(z + 1)3/2(z − 1)1/2
)
2 1
56. Solution to #6
Example
Find the derivative of y = tan(cosx).
Solution
y′ = sec2(cos x)· (− sin x) = − sec2(cos x)sin x
57. 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
58. 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))
60. Related rates of change
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
dr
A.
dA
= 2πr
dA dr
B.
dt
= 2πr +
dt
C.
dA
= 2πr
dr
dt dt
D. not enough information
Image
61. Related rates of change
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
dr
A.
dA
= 2πr
dA dr
B.
dt
= 2πr +
dt
C.
dA
= 2πr
dr
dt dt
D. not enough information
Image
62. What have we learned today?
) 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!