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3.5
Slide 3.5 - 1
The Derivatives of ax and logax
OBJECTIVES
Differentiate functions involving ax.
Differentiate functions involving logax.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3.5 The Derivatives of ax and logax
THEOREM 12
d x
a = (ln a )a x
dx
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Slide 3.5 - 3
1
3.5 The Derivatives of ax and logax
Example 1: Differentiate:
a) y = 2 x ;
b) y = (1.4)x ;
c) f (x) = 32 x.
d x
2 = (ln 2 )2 x
dx
d
x
b)
(1.4)x = (ln1.4 )(1.4 )
dx
c) f ′(x) = (ln 3)32 x ⋅ 2 = 2 ln 3⋅ 32 x
a)
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Slide 3.5 - 4
3.5 The Derivatives of ax and logax
THEOREM 13
ah − 1
h→0
h
ln a = lim
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.5 - 5
3.5 The Derivatives of ax and logax
THEOREM 14
d
1 1
log a x =
⋅
dx
ln a x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.5 - 6
2
3.5 The Derivatives of ax and logax
Example 2: Differentiate:
a) y = log 8 x;
b) y = log x;
c) f (x) = log 3 (x 2 + 1);
d) f (x) = x 3 log 5 x.
a)
d
1 1
log 8 x =
⋅
dx
ln 8 x
b)
d
1 1
log x =
⋅
dx
ln10 x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.5 - 7
3.5 The Derivatives of ax and logax
Example 2 (concluded):
c) f ′(x) =
f ′(x)
=
d) f ′(x) =
f ′(x)
=
1
1
⋅
⋅ 2x
ln 3 x 2 + 1
2x
(ln 3) x 2 + 1
(
x3 ⋅
)
1 1
⋅ + log 5 x ⋅ 3x 2
ln 5 x
x2
+ 3x 2 log 5 x
ln 5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3.5 - 8
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