12 x1 t02 01 differentiating exponentials (2014)

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Exponentials

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Page 1: 12 x1 t02 01 differentiating exponentials (2014)

Exponentials

Page 2: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

Page 3: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

Page 4: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

Page 5: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

10

aay x

Page 6: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

10

aay x

1

aay x

Page 7: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain

10

aay x

1

aay x

Page 8: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

Page 9: 12 x1 t02 01 differentiating exponentials (2014)

Exponentialsy

x

1

1

aay x

10

aay x

x real all:domain0:range y

10

aay x

1

aay x

xey

Page 10: 12 x1 t02 01 differentiating exponentials (2014)

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

Page 11: 12 x1 t02 01 differentiating exponentials (2014)

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

Page 12: 12 x1 t02 01 differentiating exponentials (2014)

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

Page 13: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

Page 14: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

Page 15: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

Page 16: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

Page 17: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

Page 18: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.

Page 19: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

Page 20: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

Page 21: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

Page 22: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii

Page 23: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii344 xe

dxdy

Page 24: 12 x1 t02 01 differentiating exponentials (2014)

Differentiating Exponentials

xfey

xfexfdxdy

xfay

xfaaxfdxdy log

xeyi e.g.xe

dxdy

xeyii 5

xedxdy 55

34 xeyiii344 xe

dxdy

232

xxeyiv

Page 25: 12 x1 t02 01 differentiating exponentials (2014)

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

Page 26: 12 x1 t02 01 differentiating exponentials (2014)

(v) xexy 423

Page 27: 12 x1 t02 01 differentiating exponentials (2014)

(v) xexy 423

xeexdxdy xx 643 442

Page 28: 12 x1 t02 01 differentiating exponentials (2014)

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

Page 29: 12 x1 t02 01 differentiating exponentials (2014)

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

Page 30: 12 x1 t02 01 differentiating exponentials (2014)

(v) xexy 423

xeexdxdy xx 643 442

126612

4

442

xxexeex

x

xx

73 2 xe yvi

xx eedxdy 363 327

Page 31: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 32: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 33: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 34: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 35: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 36: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 37: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 38: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 39: 12 x1 t02 01 differentiating exponentials (2014)

(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

Page 40: 12 x1 t02 01 differentiating exponentials (2014)

2

4 xyix

Page 41: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix

Page 42: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix x yx log

Page 43: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix x yx logyex

Page 44: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

Page 45: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

yedxdy 1

Page 46: 12 x1 t02 01 differentiating exponentials (2014)

2

44log2 xxdxdy

2

4 xyix x yx logyex ye

dydx

yedxdy 1

x1

Page 47: 12 x1 t02 01 differentiating exponentials (2014)

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