CK-12-Calculus

http://www.ck12.org Chapter 6. Transcendental Functions

d

dx[logbx] =wlim→x

logbw−logbx

w−x

=wlim→xlogwb−(wx/x)

=wlim→x

[ 1

w−xlogb

(w

x

)]

=wlim→x

[ 1

w−xlogb

(x+(w−x)

x

)]

=wlim→x

[ 1

w−xlogb

(

1 +w−xx

)]

=wlim→x

[ 1

x(w−x)logb

(

1 +w−xx

)]

=wlim→x

[ x

x(w−x)logb

(

1 +w−xx

)]

.

At this stage, leta= (w−x)/(x),the limit ofw→xthen becomesa→ 0 .Substituting, we get

=alim→ 0

[ 1

x

1

alogb(^1 +a)

]

=^1 xalim→ 0

[ 1

alogb(^1 +a)

]

=^1 xalim→ 0

[

logb( 1 +a)^1 /a

]

.

Inserting the limit,

=^1 xlogb

[

alim→ 0 (^1 +a)^1 /a

]

.

But by the definitione=lima→ 0 ( 1 +a)^1 /a,

d

dx[logbx] =

1

xlogbe.

From the box above, we can express logbein terms of natural logarithm by the using the formula logbw=lnw/lnb.
Then

logbe=lnlneb=ln^1 b.

Thus we conclude

d

dx[logbx] =

1

xlnb>^0 ,