Differentiating Trig

Differentiating Trig0sinlim

0

x

x

Differentiating Trig0sinlim

0

x

x

1coslim0

xx

Differentiating Trig0sinlim

0

x

x

1coslim0

xx

0tanlim0

xx

Differentiating Trig0sinlim

0

x

x Sector Area OAC

1coslim0

xx

0tanlim0

xx

O A

C

1x

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea

1coslim0

xx

0tanlim0

xx

O A

C

1x

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

1coslim0

xx

0tanlim0

xx

O A

C

1x

B

tan

x

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2 1coslim

0

x

x

0tanlim0

xx

O A

C

1x

B

tan

x

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2 1coslim

0

x

x

0tanlim0

xx

O A

C

1x

B

tan

x

xxx tansin

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2 1coslim

0

x

x

0tanlim0

xx

O A

C

1x

B

tan

x

xxx tansin

xx

xx

xx

sintan

sinsinsin

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2 1coslim

0

x

x

0tanlim0

xx

O A

C

1x

B

tan

x

xxx tansin

xx

xx

xx

sintan

sinsinsin

xxx

cos1

sin1

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2 1coslim

0

x

x

0tanlim0

xx

O A

C

1x

B

tan

x

xxx tansin

xx

xx

xx

sintan

sinsinsin

xxx

cos1

sin1

1sin

1

0as

xx

x

Differentiating Trig0sinlim

0

x

x Sector Area OACAOCArea AOB Area

xxx tan1211

21sin11

21 2

1sin

lim0

x

xx

1coslim0

xx

0tanlim0

xx

O A

C

1x

B

tan

x

xxx tansin

xx

xx

xx

sintan

sinsinsin

xxx

cos1

sin1

1sin

1

0as

xx

x

e.g. x

xix 5sin

5lim0

e.g. x

xix 5sin

5lim0

1

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0 xx

x 3sin3

31lim

0

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0 xx

x 3sin3

31lim

0

31

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sinx

xx 3sin

331lim

0

31

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

hxxx

x

sinsinhcoscoshsinlim0

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

hxxx

x

sinsinhcoscoshsinlim0

hx

hx

x

1coshsinsinhcoslim0

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

hxxx

x

sinsinhcoscoshsinlim0

hx

hx

x

1coshsinsinhcoslim0

h

h

xh

xx

22

cos2sinsinhcoslim

2

0

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

hxxx

x

sinsinhcoscoshsinlim0

hx

hx

x

1coshsinsinhcoslim0

h

h

xh

xx

22

cos2sinsinhcoslim

2

0 1cos22cos 2

e.g. x

xix 5sin

5lim0

1 x

xiix 3sinlim

0

xy sin

h

xhxdxdy

x

sinsinlim0

xx

x 3sin3

31lim

0

31

hxxx

x

sinsinhcoscoshsinlim0

hx

hx

x

1coshsinsinhcoslim0

h

h

xh

xx

22

cos2sinsinhcoslim

2

0 1cos22cos 2

h

h

xh

xx

2sin2

sinsinhcoslim2

0

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx xfy sin

xfxfdxdy cos

xcos

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

xy

2sin

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

xy

2sin

x

dxdy

2cos

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

xy

2sin

x

dxdy

2cos

xsin

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

xy

2sin

x

dxdy

2cos

xsin

xfy cos

2

2sin

sinsinhcoslim2

0 h

h

xh

xx

2sin

2

2sin

sinsinhcoslim0

hh

h

xh

xx

0sin1cos xx

xcos

xfy sin

xfxfdxdy cos

xy cos

xy

2sin

x

dxdy

2cos

xsin

xfy cos

xfxfdxdy sin

xy tan

xy tan

xxy

cossin

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xtan

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xtan

xyiv 5tan

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xtan

xyiv 5tan

xxdxdy 24 sectan5

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xtan

xyiv 5tan

xxdxdy 24 sectan5

xeyv cos

xy tan

xxy

cossin

xxx

xxxxx

dxdy

2

22

2

cossincos

cossinsincoscos

xx

2

2

seccos

1

xfy tan

xfxfdxdy 2sec

3sin e.g. xyi 32 cos3 xx

dxdy

x

yii 1tan

xxdxdy 1sec1 2

2

xyiii coslog

xx

dxdy

cossin

xtan

xyiv 5tan

xxdxdy 24 sectan5

xeyv cos xx ee

dxdy sin

Exercise 14F; 2, 4, 5

Exercise 14G; 2ace etc, 3ace etc, 5ace etc, 6, 7ab(i), 8, 12, 13c, 16a*

Exercise 14H; 2 ac, 4bd, 6, 8, 14, 17, 20