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