12X1 T02 01 differentiating exponentials (2011)

Exponentials

Exponentialsy

x

Exponentialsy

x

1

1

aay x

Exponentialsy

x

1

1

aay x

10

aay x

Exponentialsy

x

1

1

aay x

10

aay x

10

aay x

Exponentialsy

x

1

1

aay x

10

aay x

10

aay x

1

aay x

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain

10

aay x

1

aay x

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

xey

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

xey

e is an irrational number, it is defined as; n

n

n

11lim

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

xey

e is an irrational number, it is defined as; n

n

n

11lim

718281828.2e

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

xey

e is an irrational number, it is defined as; n

n

n

11lim

718281828.2edp) 3 (to 086.20 e.g. 3 e

Differentiating Exponentials

Differentiating Exponentials

xfey

Differentiating Exponentials

xfey

xfexfdxdy

Differentiating Exponentials

xfey

xfexfdxdy

xfay

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii344 xe

dxdy

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii344 xe

dxdy

232

xxeyiv

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii344 xe

dxdy

232

xxeyiv

232

32 xxexdxdy

(v) xexy 423

(v) xexy 423

xeexdxdy xx 643 442

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

1,1point at the

1 o tangent t theFind 2

2

eeyviii x

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

1,1point at the

1 o tangent t theFind 2

2

eeyviii x

x

x

edxdy

ey

2

2

2

1

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

1,1point at the

1 o tangent t theFind 2

2

eeyviii x

x

x

edxdy

ey

2

2

2

1

22,1when edxdyx

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

1,1point at the

1 o tangent t theFind 2

2

eeyviii x

x

x

edxdy

ey

2

2

2

1

22,1when edxdyx

121 22 xeey

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

633 221 xx ee

3

x

x

eeyvii

23

3

x

xxxx

eeeee

dxdy

2

2

22

33

33

x

x

x

xxx

eee

eee

1,1point at the

1 o tangent t theFind 2

2

eeyviii x

x

x

edxdy

ey

2

2

2

1

22,1when edxdyx

121 22 xeey

012221

22

222

eyxeexeey

2

4 xyix

2

44log2 xxdxdy

2

4 xyix

2

44log2 xxdxdy

2

4 xyix x yx log

2

44log2 xxdxdy

2

4 xyix x yx logyex

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

yedxdy 1

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

yedxdy 1

x1

2

44log2 xxdxdy

Exercise 13A; 1 to 4 ace etc, 6 to 8 ace, 10 to 12 ac

Exercise 13B; 4, 7, 8 to 22 evens (not 18)

2

4 xyix x yx logyex ye

dydx

yedxdy 1

x1