5.4Exponential Functions:
Differentiation and Integration
The inverse of f(x) = ln x is f-1 = ex.
Therefore, ln (ex) = x and e ln x = x
Solve for x in the following equations.
17 += xe Take the ln of both sides.
1ln7ln += xe17ln +=x
95.17ln ≈−=x
5)32ln( =−x5)32ln( ee x =−
532 ex =−707.75
2
3 5
≈+
=e
x
Operations with Exponential Functions
baba eee += bab
a
ee
e −=
€
ea( )b
= eab
The Derivative of the Natural Exponential Function
€
d
dxex[ ] = ex
€
d
dxeu[ ] = euu'
Differentiate.
€
d
dxe2x−1
[ ] = 122 −xe
€
d
dxe−3 x
[ ] = xex
32
3 −⎟⎠
⎞⎜⎝
⎛
Find the relative extrema of xxexf =)(
)1()()(' xx eexxf +=( )10 += xex Since ex never = 0, -1 is the only
critical number.
-1
neg.
dec.
pos.
inc.
Therefore, x = -1 is a min. bythe first derivative test.
Minimum @ ? ⎟⎠
⎞⎜⎝
⎛ −−e
1,1
Integration Rules for Exponential Functions
CedueCedxe uuxx +=+= ∫∫Ex.
dxe x∫ +13Let u = 3x + 1
du = 3 dx
dxdu
=3
3
dueu∫=
Ceu
+=3
Ce x
+=+
3
13
Ex. dxxe x∫ − 2
5 Let u = -x2
du = -2x dx
dxx
du=
−2x
duxeu
25
−=∫
Ceu +−=25
Ce x +−= − 2
25
Ex.
dxx
e x
∫ 2
1Let u = 1/x = x-1
dxxdu 21 −−=
dxx
du=
− −2
dxdux =− 2
)( 2
2dux
x
eu −=∫Ceu +−=Ce x +−= 1
Ex. dxex x∫ ⋅ cossin Let u = cos x
du = -sin x dx
dxx
du=
−sinx
duex u
sinsin
−⋅=∫
CeCe xu +−=+−= cos
Ex.dxe x∫ −
1
0
Let u = -x du = -dx -du = dx
( )dueu −= ∫
€
=−eu = −e−x]
0
1
( )01 ee −−−= − 632.1
1 ≈−=e
Ex. dxe
ex
x
∫ +
1
0 1 u
u'=
€
=ln 1+ ex( )]0
1
( ) 2ln1ln −+= e
620.≈Ex.
€
ex cos ex( )[ ]−1
0
∫ dx Let u = ex
du = ex dx
dxe
dux=
€
= ex cos u( )[ ]∫ du
ex
€
=sinu = sin(ex )]−1
0
482.)sin(1sin 1 ≈−= −e
