Transcript 2.2 Basic Differentiation Rules and Rates of Change Objective: Find

Ms. Battaglia
AB/BC Calculus
Thm 2.2 The Constant Rule
The derivative of a constant function is
0. That is, if c is a real number, then
𝑑
𝑐 =0
𝑑𝑥
Example:
Function
Derivative
a. y = 10
dy/dx =
b. f(x) = 0
f’(x) =
c. s(t) = -2
s’(t) =
d. y = k𝜋 2 , k is constant
y’ =
Thm 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 to be differentiable at x=0, n must be a number
such that 𝑥 𝑛−1 is defined on an interval containing 0.
a. 𝑓 𝑥 = 𝑥
4
b. 𝑔 𝑥 =
6
𝑥
c. 𝑦 =
1
𝑥8
Find the slope of the graph of 𝑓 𝑥 = 𝑥 1/2 when
a. x = 0
b. x = 1
c. x = 4

Find an equation of the tangent line
to the graph of 𝑓 𝑥 = 𝑥 −1 when x=1.
Thm 2.4 The Constant Multiple Rule
If f is a differentiable function and c is a real number,
then cf is also differentiable and
𝑑
𝑑𝑥
𝑐𝑓 𝑥
= 𝑐𝑓′(𝑥).
Thm 2.5 The Sum and Difference Rules
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.
𝑑
𝑓 𝑥 + 𝑔(𝑥) = 𝑓 ′ 𝑥 + g′(x)
𝑑𝑥
𝑑
𝑓 𝑥 − 𝑔(𝑥) = 𝑓 ′ 𝑥 − g′(x)
𝑑𝑥
a. 𝑦 =
4
𝑥
b. 𝑓 𝑡 =
2𝑡 3
5
c. 𝑦 = 6 𝑥
d. 𝑦 =
2
3
3
𝑥2
e. y=
2𝑥
−
7
Original
Function
Rewrite
Differentiate
Simplify
5
𝑦= 3
2𝑥
5 −3
𝑦 = (𝑥 )
2
5
𝑦′ = (−3𝑥 −4 )
2
15
𝑦= 4
2𝑥
𝑦=
5
(2𝑥)3
7
𝑦 = −2
3𝑥
7
𝑦=
(3𝑥)−2
a. 𝑓 𝑥 =
3𝑥 2
−𝑥+𝜋
b. 𝑔 𝑥 =
𝑥2
−
2
+ 𝑥 3 − 8𝑥
Theorem 2.6
𝑑
𝑑𝑥
sin𝑥 = cos𝑥
𝑑
𝑑𝑥
cos𝑥 = −sin𝑥
a. 𝑦 = 2sin𝑥 + 7
b. 𝑦 =
2sin𝑥
3
c. 𝑦 = 𝑥 − cos𝑥

Determine the point(s) (if any) at which the graph
of 𝑦 = 𝑥 3 + 𝑥 has a horizontal tangent line.
distance
Rate =
time
the average velocity is
Change in distance ∆s
=
Change in time
∆t
 AB:
Page 116 #59-65 odd, 79,
107, 110, 111, 113, 117,119,
graphing worksheet
 BC:
Page 116 #59-65 odd, 79,
107, 110, 111, 113, 117,119, AP
sample problem worksheet