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