Warm up
— Product Rule

— Quotient Rule

1

Power Rule n f (x) = x f '(x) = nx

n−1

2

Power Rule n f (x) = x f '(x) = nx

n−1

General Power Rule n f (x) = g(x) n−1 f '(x) = ng(x) g"(x)
3

Examples y = ( 2x + 3)

3

y = x 3 −1

4

Exercise A
General Power Rule

5

Transcendental Functions x sin(x)

e

cos(x)

ln(x)

6

Sine Function
— ratio of the length of the side opposite the angle to the hypotenuse

hyp

opp

θ opp sin(θ ) = hyp 7

Radians vs Degrees
Radians

Degrees

0

0

π/2

90

π

180

3π/2

270

2π

360

8

Plot of sin(x)

−2π − 3π
2

−π

−

π
2

π
2

π

3π
2

2π

9

Values of sin(x)
Radians

Degrees

sin(0)=0

sin(0)=0

sin(π/2)=1

sin(90)=1

sin(π)=0

sin(180)=0

sin(3π/2)=-1

sin(270)=-1

sin(2π)=0

sin(360)=0

10

Cosine Function
— ratio of the length of the side adjacent the angle to the hypotenuse

hyp

θ adj adj cos(θ ) = hyp 11

Plot of cos(x)

−2π

3π
−
2

−π

−

π
2

π
2

π

3π
2

2π

12

Values of cos(x)
−2π

3π
−
2

−π

−

π
2

π
2

π

3π
2

2π

Radians

Degrees

cos(0)=

cos(0)=

cos(π/2)=

cos(90)=

cos(π)=

cos(180)=

cos(3π/2)=

cos(270)=

cos(2π)=

cos(360)=

13

Exercise B sin and cos

14

Derivative of sin(x)

−2π

−2π

−π

−π

π
−
2

π
−
2

π
2

π
2

π

π

3π
2

2π

3π
2

2π

15

Derivative of sin(x)

−2π

−π

π
−
2

π
2

π

3π
2

2π

3π
2

2π

=cos(x)
−2π

−π

π
−
2

π
2

π

16

Proof y = sin(x)

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

f (x + h) − f (x) y' = lim h→0 h sin(x + h) − sin(x) y' = lim h→0 h
[sin(x)cos(h) + cos(x)sin(h)]− sin(x) y' = lim h→0 h