2-2: Basic Differentiation Rules
Objectives:
1. To derive and use
the basic rules of
differentiation
Assignment:
• P. 115-118: 1, 3-23
odd, 27, 29, 33, 39-61
eoo, 63, 69, 70, 83-89,
92, 111, 113, 115, 116
Warm-Up
Use the limit definition to find the derivative
of 𝑦 = 𝑐, where 𝑐 ∈ ℝ.
Constant Rule
Power Rule
Constant Multiple Rule
Objective 1
You will be able to derive and
use the basic rules of
differentiation
Sum/Difference Rule
Sine/Cosine Rules
Local Linearity
𝑦 = 𝑥2
𝑦 = 𝑥3
𝑦 = 𝑥4
Derivative
Function
𝑦=𝑥
⋯
𝑦′ = 1
𝑦′ = 2𝑥
𝑦′ = 3𝑥 2
𝑦′ = 4𝑥 3
⋯
Pascal’s Triangle
𝑥+𝑐
𝑥+𝑐
4
2
𝑥 2 + 2𝑥𝑐 + 𝑐 2
𝑥 4 + 4𝑥 3 𝑐 + 6𝑥 2 𝑐 2 + 4𝑥𝑐 3 + 𝑐 4
Exercise 1
Use Pascal’s Triangle to expand 𝑥 + 𝑐 3 .
Binomial Theorem
In the expansion of 𝑥 + 𝑐
𝑥+𝑐
𝑛
𝑛
= 𝑥 + 𝑛𝑥
𝑛−1
𝑛
𝑛!
𝑐 + ⋯+
𝑥 𝑛−𝑟 𝑐 𝑟 + ⋯ + 𝑛𝑥𝑐 𝑛−1 + 𝑐 𝑛
𝑛 − 𝑟 ! 𝑟!
This is the number of
combinations of 𝑛
objects taken 𝑟 at a time.
Pascal’s Triangle
For 𝑟 = 4:
10!
=
10 − 4 ! 4!
10!
=
6! 4!
10 ∙ 9 ∙ 8 ∙ 7 ∙ 6!
=
6! 4 ∙ 3 ∙ 2 ∙ 1
210
Exercise 2
Show that
𝑑
𝑑𝑥
𝑥 𝑛 = 𝑛𝑥 𝑛−1 .
Constant and Power Rules
Constant Rule
Power Rule
𝑑
𝑐 =0
𝑑𝑥
𝑑 𝑛
𝑥 = 𝑛𝑥 𝑛−1
𝑑𝑥
𝑐, 𝑛 ∈ ℝ
Exercise 3a
Find the derivative of the following.
1. 𝑦 = 7
2. 𝑠 𝑡 = −3
3.
𝑑
𝑑𝑥
𝑘𝜋 2
Exercise 3b
Find the derivative of the following.
1. 𝑓 𝑥 = 𝑥 3
2. 𝑔 𝑥 =
3
𝑥
3. 𝑦 =
1
𝑥2
Exercise 4
Find an equation of the tangent line to the
graph of 𝑓(𝑥) = 𝑥 2 when 𝑥 = −2.
Exercise 5
Find
𝑑
𝑑𝑥
𝑐𝑓(𝑥) .
Constant Multiple Rule
Constant Multiple Rule
𝑑
𝑐𝑓(𝑥) = 𝑐𝑓′(𝑥)
𝑑𝑥
𝑐∈ℝ
Exercise 6
Find the derivative of the following.
1. 𝑦 =
2
𝑥
2. 𝑓 𝑡 =
4𝑡 2
5
3. 𝑦 =
1
3
2 𝑥2
Exercise 7
Find
𝑑
𝑑𝑥
𝑓 𝑥 +𝑔 𝑥 .
Sum and Difference Rule
The sum or difference of two differentiable
functions 𝑓 and 𝑔 is itself differentiable.
Moreover, the derivative of 𝑓 + 𝑔 or 𝑓 − 𝑔
is the sum or difference of the derivatives
of 𝑓 and 𝑔.
Sum Rule
Difference Rule
𝑑
𝑓 𝑥 + 𝑔(𝑥) = 𝑓 ′ 𝑥 + 𝑔′(𝑥)
𝑑𝑥
𝑑
𝑓 𝑥 − 𝑔(𝑥) = 𝑓 ′ 𝑥 − 𝑔′(𝑥)
𝑑𝑥
Exercise 8
Find the derivative of the following.
1. 𝑓 𝑥 = 𝑥 3 − 4𝑥 + 5
2. 𝑔 𝑥 =
𝑥4
−
2
+ 3𝑥 3 − 2𝑥
Exercise 9
Use local linearity to approximate the
derivative of 𝑓(𝑥) = sin 𝑥.
Sine and Cosine Rules
𝑑
sin 𝑥 = cos 𝑥
𝑑𝑥
𝑑
cos 𝑥 = − sin 𝑥
𝑑𝑥
Exercise 10
Show that
𝑑
𝑑𝑥
sin 𝑥 = cos 𝑥.
Exercise 11
Find the derivative of the following.
1. 𝑦 =
sin 𝑥
2
2. 𝑦 = 𝑥 + cos 𝑥
2-2: Basic Differentiation Rules
Objectives:
1. To derive and use
the basic rules of
differentiation
Assignment:
• P. 115-118: 1, 3-23
odd, 27, 29, 33, 39-61
eoo, 63, 69, 70, 83-89,
92, 111, 113, 115, 116