2-3: Product and Quotient Rules
Objectives:
Assignment:
1. To derive and use the
Product and Quotient
Rules
• P. 126-129: 1-11 odd, 15,
17, 25, 31, 35, 37, 47, 53,
66, 81, 87, 129, 131
2. To find higher-order
derivatives
• P. 128-129: 93-101 odd,
109, 111, 130, 132, 133
Warm Up 1
Recall that the Sum Rule
states that the derivative of
a sum is the sum of the
derivatives. Is the same
true for products?
Let 𝑓(𝑥) = (𝑥 2 + 1)(𝑥 − 1).
Find 𝑓′(𝑥).
Warm Up 2
Recall that the Sum Rule
states that the derivative of
a sum is the sum of the
derivatives. Is the same
true for products?
Let 𝑓(𝑥) = (𝑥 2 + 1)(𝑥 − 1).
Expand 𝑓(𝑥), then find 𝑓′(𝑥).
Objective 1
You will be able to derive
and use the Product and
Quotient Rules
Exercise 1
Let 𝑓(𝑥) and 𝑔(𝑥) be differentiable functions.
Use the limit definition of derivative to find
𝑑
𝑓(𝑥) ∙ 𝑔(𝑥) .
𝑑𝑥
Product Rule
The product of two differentiable functions 𝑓
and 𝑔 is also differentiable such that
𝑑
𝑓(𝑥) ∙ 𝑔(𝑥) = 𝑓 ′ 𝑥 ∙ 𝑔 𝑥 + 𝑓(𝑥) ∙ 𝑔′ (𝑥)
𝑑𝑥
Derivative of the first times the second + first times the derivative of the second
Exercise 2
Let 𝑓(𝑥) = (𝑥 2 + 1)(𝑥 − 1). Find 𝑓′(𝑥).
Exercise 3
Find the derivative of 𝑦 = 3𝑥 2 sin 𝑥
Exercise 4
Find the derivative of 𝑦 = 2𝑥 cos 𝑥 − 2 sin 𝑥
Exercise 5
Let 𝑦 = 𝑥 sin 𝑥 cos 𝑥 . Find 𝑑𝑦/𝑑𝑥.
Product Rule
The product of three differentiable functions
𝑓, 𝑔, and ℎ is also differentiable such that
𝑑
𝑓(𝑥) ∙ 𝑔(𝑥) ∙ ℎ 𝑥
𝑑𝑥
= 𝑓 ′ 𝑥 ∙ 𝑔 𝑥 ∙ ℎ 𝑥 + 𝑓 𝑥 ∙ 𝑔′ 𝑥 ∙ ℎ 𝑥 + 𝑓 𝑥 ∙ 𝑔 𝑥 ∙ ℎ′ 𝑥
Exercise 6
Recall that the Difference
Rule states that the
derivative of a difference is
the difference of the
derivatives. Is the same
true for quotients?
Let 𝑓(𝑥) =
𝑥 2 −1
.
𝑥+1
Find 𝑓′(𝑥).
Exercise 7
Let 𝑓(𝑥) and 𝑔(𝑥) be differentiable functions
for all 𝑥 such that 𝑔(𝑥) ≠ 0. Use the limit
𝑑 𝑓(𝑥)
definition of derivative to find
.
𝑑𝑥 𝑔(𝑥)
Quotient Rule
The quotient of two differentiable functions 𝑓
and 𝑔 is also differentiable for all 𝑥 where
𝑔(𝑥) ≠ 0 such that
𝑑 𝑓(𝑥)
𝑔 𝑥 ∙ 𝑓 ′ 𝑥 − 𝑓(𝑥) ∙ 𝑔′(𝑥)
=
𝑑𝑥 𝑔(𝑥)
𝑔 𝑥 2
The bottom times the derivative of the top minus the top times the
derivative of the bottom all divided by the bottom squared
Exercise 8
Find the derivative of 𝑓(𝑥) =
𝑥 2 −1
𝑥+1
Exercise 9
Find the equation of line that is tangent to
𝑦=
1
3−
𝑥
𝑥+5
at the point (−1,1).
Try rewriting as a
single fraction
Pay attention to
parenthesis when
differentiating
9
𝑦= 2
5𝑥
−3 3𝑥 − 2𝑥 2
𝑦=
7𝑥
5𝑥 4
𝑦=
8
𝑥 + require
1
None of these
=
the𝑦Quotient
𝑥 − 1Rule
𝑥 2 + 3𝑥
𝑦=
6
Exercise 10
Which of the following does not belong?
Exercise 11
Find the derivative of each function.
1. 𝑦 = tan 𝑥
2. 𝑦 = sec 𝑥
Trigonometric Derivatives
𝑑
tan 𝑥 = sec 2 𝑥
𝑑𝑥
𝑑
cot 𝑥 = − csc 2 𝑥
𝑑𝑥
𝑑
sec 𝑥 = sec 𝑥 tan 𝑥
𝑑𝑥
𝑑
csc 𝑥 = − csc 𝑥 cot 𝑥
𝑑𝑥
Exercise 12
Differentiate both forms of
1 − cos 𝑥
𝑦=
= csc 𝑥 − cot 𝑥
sin 𝑥
Objective 2
You will be able to find
higher-order derivatives
Exercise 13
Let 𝑓(𝑥) = 𝑥 2 + 3𝑥 − 5. Find the
instantaneous rate of change at 𝑥 = 𝑐. What
is the rate of change of the instantaneous
rate of change?
Higher-Order Derivatives
The derivative of a function gives the rate of
change of that function, which could also be
changing…
∆ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
∆ 𝑡𝑖𝑚𝑒
∆ 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
∆ 𝑡𝑖𝑚𝑒
∆ 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑡𝑖𝑜𝑛
∆ 𝑡𝑖𝑚𝑒
∆ 𝑗𝑒𝑟𝑘
∆ 𝑡𝑖𝑚𝑒
1st derivative
2nd derivative
3rd derivative
4th derivative
Velocity
Acceleration
Jerk
Jounce
Higher-Order Derivatives
The derivative of the first derivative is called the second
derivative. The derivative of the second derivative is
the third derivative…
1st derivative
𝑦′
𝑓′(𝑥)
2nd derivative
𝑦′′
𝑓′′(𝑥)
3rd derivative
𝑦′′′
𝑓′′′(𝑥)
4th
derivative
𝑦 (4)
𝑓
4
derivative
𝑦 (𝑛)
𝑓
𝑛
𝑛th
(𝑥)
(𝑥)
𝑑𝑦
𝑑𝑥
𝑑2𝑦
𝑑𝑥 2
𝑑
𝑓(𝑥)
𝑑𝑥
𝑑2
𝑓(𝑥)
𝑑𝑥 2
𝑑3𝑦
𝑑𝑥 3
𝑑4𝑦
𝑑𝑥 4
𝑑𝑛 𝑦
𝑑𝑥 𝑛
𝑑3
𝑓(𝑥)
𝑑𝑥 3
𝑑4
𝑓(𝑥)
𝑑𝑥 4
𝑑𝑛
𝑓(𝑥)
𝑑𝑥 𝑛
𝐷𝑥 𝑦
𝐷𝑥2 𝑦
𝐷𝑥3 𝑦
𝐷𝑥4 𝑦
𝐷𝑥𝑛 𝑦
Exercise 14
Find the second derivative of 𝑦 = 𝑥 sin 𝑥.
Exercise 15
Label each
function
as 𝑓, 𝑓′,
𝑓′′, and
𝑓′′′.
Exercise 15
Label each
function
as 𝑓, 𝑓′,
𝑓′′, and
𝑓′′′.
2-3: Product and Quotient Rules
Objectives:
Assignment:
1. To derive and use the
Product and Quotient
Rules
• P. 126-129: 1-11 odd, 15,
17, 25, 31, 35, 37, 47, 53,
66, 81, 87, 129, 131
2. To find higher-order
derivatives
• P. 128-129: 93-101 odd,
109, 111, 130, 132, 133