Download 2.2 Basic Differentiation Rules and Rates of Change

2.2 Basic
Differentiation Rules
and Rates of Change
AP Calculus AB
Objectives: Find the derivative of a
function using rules
Please pick-up a clicker. They will be used at the end.
Theorem 2.2
The Constant Rule
• The derivative of a constant is zero
𝑑
𝑐 =0
𝑐 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟
𝑑𝑥
• What is the rate of change of a constant function?
Theorem 2.3
The Power Rule
• If n is a rational number, then the function 𝑓 𝑥 = 𝑥 𝑛
is differentiable and
𝑑 𝑛
𝑥 = 𝑛𝑥 𝑛−1
𝑑𝑥
• For f(x) to be differentiable at x = 0, n must be a
number such that 𝑥 𝑛−1 is defined on an interval
containing 0.
Calculus-Help
• http://www.calculus-help.com/tutorials/
Examples
1. 𝑓 𝑥 = 𝑥 3
𝑓′ 𝑥 =
2. 𝑔 𝑥 = 3 𝑥
𝑔′ 𝑥 =
.
𝑔 𝑥 =𝑥
1
3. 𝑦 = 𝑥 2
.
𝑦 = 𝑥 −2
1
3
𝑦′ =
Find the Slope at a Point
• Find the slope of 𝑦 = 𝑥 4 when x = -1, 0, 1
• What should we do first?
𝑦 ′ = 4𝑥 3
• Then we can plug in the x-values to find the slope of
a tangent line
• 𝑦 ′ −1 = 4(−1)3 = −4
• 𝑦′ 0 = 4 0 3 = 0
• 𝑦′ 1 = 4 1 3 = 4
Find the Equation of the
Tangent
• Find the equation of the tangent line to 𝑓 𝑥 = 𝑥 2
when x = -2.
𝑓 ′ 𝑥 = 2𝑥
𝑓 ′ −2 = 2 −2 = −4
𝑓 −2 = (−2)2
𝑚 = −4,
−2, 4
𝑦 − 4 = −4(𝑥 + 2)
Theorem 2.4
The Constant Multiple Rule
• If f(x) is a differentiable function and c is a real
number, then cf(x) is also differentiable and
𝑑
𝑐𝑓 𝑥 = 𝑐𝑓′(𝑥).
𝑑𝑥
Examples
1. 𝑓 𝑥 = 4𝑥 2
2
𝑦′ =
2. 𝑦 = 𝑥
.
𝑦 = 2𝑥 −1
3. 𝑓 𝑡
.
𝑓 𝑡
4. 𝑦 =
.
𝑓′ 𝑥 =
𝑦=
4𝑡 2
=
5
4
= 𝑡2
5
1
3
2 𝑥2
1 −2
𝑥 3
2
𝑓 ′ (𝑡) =
𝑦′ =
Constant Multiple and
Power Rule Together
𝑑
𝑐𝑥 𝑛 = 𝑐𝑛𝑥 𝑛−1
𝑑𝑥
Theorem 2.5
The Sum and Difference Rules
𝑑
𝑓 𝑥 ±𝑔 𝑥
𝑑𝑥
= 𝑓′(𝑥) ± 𝑔′(𝑥)
When adding or subtracting functions together, we
must find the derivative of each part separately and
continue to add or subtract those parts.
Examples
1. 𝑦 = 𝑥 3 − 4𝑥 + 5
.
𝑦′ =
1
2. 𝑔 𝑥 = − 2 𝑥 4 + 3𝑥 3 − 2𝑥
.
𝑔′ 𝑥 =
Theorem 2.6
Derivatives of Sine and Cosine
•
𝑑
(sin 𝑥)
𝑑𝑥
= cos 𝑥
•
𝑑
(cos 𝑥)
𝑑𝑥
= − sin 𝑥
• YOU MUST KNOW THESE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
Examples
1. 𝑦 = 2 sin 𝑥
2. 𝑓 𝑥 =
.
𝑓 𝑥 =
sin 𝑥
2
1
sin 𝑥
2
𝑦′ =
𝑓′ 𝑥 =
3. 𝑔 𝑥 = 𝑥 + cos 𝑥
𝑔′ 𝑥 =
4. ℎ 𝑥 = 5 + cos 𝑥
ℎ′ 𝑥 =
Formative Assessment
• Pg. 115 (2-38) even
• Exit Question
• Join Code 13
• What is the derivative of 𝑦 = −9𝑥 3 + 3𝑥 2 − 2𝑥 + 9?
(A) 𝑦 ′ = −9𝑥 2 + 3𝑥 − 2
(C) 𝑦 ′ = −27𝑥 2 + 6𝑥 + 2
(E) 𝑦 ′ = −27𝑥 2 + 6𝑥 − 2
(B) 𝑦 ′ = −27𝑥 3 + 6𝑥 2 − 2𝑥 + 9
(D) 𝑦 ′ = −27𝑥 2 − 6𝑥 − 2