chapter3: differentiation derivatives of trigonometric functions: chain rule: implicit diff....

Chapter3: Differentiation

xxdx

dcos)(sin

xxdx

dsin)(cos

xxdx

d 2sec)(tan

xxxdx

dcotcsc)(csc

xxxdx

dtansec)(sec

xxdx

d 2csc)(cot

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS:

)(')(')(' xgxgfxH Chain Rule:

Implicit diff.

))(('

1' )(

11

xffxf

axg

xgxg

dx

da ln

1

)(

)(')(log

)(

)(')(ln

xg

xgxg

dx

d

x

xx '

)(')()()(')()( xgxfxgxfxgxfdx

d

Derivative Product Rule

2)]([

)(')()()('

)(

)(

xg

xgxfxgxf

xg

xf

dx

d

Derivative Quotient Rule Derivative of exponential

)(ln)( aaadx

d xx

0a

ax

xf

at

diff )(

ax

xf

at

cont )(

exist

)(lim xfax

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) others

Exam-2 Problems

)2()(kf

Chapter3: Differentiation

The slope of the tangent line

)2(' fFind

The slope of the tangent line The rate of change

Exam-2 Problems

)9('y

)0('/1 f )8/(' s

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Sec 3.7: Implicit Differentiation

Sec 3.7: Implicit Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Sec 3.7: Implicit Differentiation

Sec 3.7: Implicit Differentiation

Sec 3.7: Implicit Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) others

Exam-2 Problems

)2()(kf

(Sketch and find the slopes)

Sec 3.9: Inverse Trigonometric Functions

Sec 3.9: Inverse Trigonometric Functions

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) others

Exam-2 Problems

)2()(kf

Chapter3: Differentiation

)(

)(lim

xg

xfax 0

0

)('

)('lim

)(

)(lim

xg

xf

xg

xfaxax

Chapter3: Differentiation

Sec 3.5: Derivatives Of Trigonometric Functions

Sec 3.5: Derivatives Of Trigonometric Functions

Sec 3.5: Derivatives Of Trigonometric Functions

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) others

Exam-2 Problems

)2()(kf

Sec 3.5: Derivatives Of Trigonometric Functions

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) others

Exam-2 Problems

)2()(kf

Chapter3: Differentiation

Chapter3: Differentiation

TERM-121 Exam-2

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

1) Direct: Find the g’(2)

2) Indirect: Find the slope….

3) Find slope in different ways

4) Find

5) Find: lim

6) Find f’(x)

7) Others 1) linearization, 2) differential,

3) Related 4) rate, rate ofchange,

5) cont+diff,

Exam-2 Problems

)2()(kf

TERM-121 Exam-2

Chapter3: Differentiation

TERM-121 Exam-2

Chapter3: Differentiation

Chapter3: Differentiation

TERM-121 Exam-2

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Chapter3: Differentiation

Sec 3.11: Linearization and Differentials

Sec 3.11: Linearization and Differentials

Sec 3.11: Linearization and Differentials

Exa

mp

le:

F09

1

Sec 3.11: Linearization and Differentials

Example: F091

Sec 3.11: Linearization and Differentials

Example: F091

Sec 3.11: Linearization and Differentials