AP CALCULUS

1009: TRIG FUNCTIONS and e

Derivative of Sine -Graphically

sin( )y x

02

3

2

2

m m

m m

m

cos( )? ?y x

cos( )y x

Derivative of Sine - Analytically

REM: and 0

sin( )lim 1x

x

x

0

cos( ) 1lim 0x

x

x

sin( )d

xdx

Derivative of Cosine - Analytically

REM: and 0

sin( )lim 1x

x

x

0

cos( ) 1lim 0x

x

x

cos( )d

xdx

TRIG DERIVATIVES

TRIG Quotients and Reciprocal Identities:

tand

xdx

Likewise (cot )d

xdx

sin

cos

d x

dx x

TRIG DERIVATIVES

TRIG Quotients and Reciprocal Identities:

1(csc )

sin

d dx

dx dx x

Likewise dsec

dxx

TRIG DERIVATIVES

TRIG: Make List of Trig Derivatives …Memorize!

sin( ) cos( )

tan( ) cot( )

sec( ) csc( )

y x y x

y x y x

y x y x

Derivative of - Analytically

REM: 0

1lim 1

x

x

e

x

xde

dx

xe

Derivative of - Analytically𝑦=ln (𝑥)

y=lim𝑥→ 0

ln (𝑥)

y=ln (𝑥)

𝑒𝑦=𝑥

𝑒𝑦 𝑑𝑦𝑑𝑥

=1

𝑑𝑦𝑑𝑥

=1

𝑒𝑦=1𝑥

Therefore the derivative of

𝑦=ln𝑥

𝑦 ′=1𝑥

Think about

𝑦 ′=1𝑢∗𝑢 ′

y=ln (2𝑥+3)

𝑦 ′=1

2𝑥+3∗2

Last Update• 9/21/11

Assignment: p. 146 # 1 , 5 , 9 , 15 , 17 , 23