Calculus Section 3.9b
Page51
3.9b
More on Derivatives of Exponential and
Logarithmic Functions
Yesterday you learned several new formulas
e
x
( )
′
=
e
u
( )
′
=
ln
x
(
)
′
=
ln
u
(
)
′
=
The natural logarithm function obeys all of the laws of logarithms that you learned in
Mathematics 12.
ln
x
=
y
⇔
ln1
=
ln
e
=
ln
e
x
=
e
ln
x
=
ln
ab
=
ln
a
b
=
ln
a
n
=
log
b
a
=
Logarithmic Differentiation
(technique of taking the ln of both sides then differentiating implicitly)
Example 1
Find the derivative of
y
=
x
x
Example 2
Find the derivative of
y
=
3
x
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Page 52
We now have a new “shortcut” formula for taking the derivative of a function in the
form
y
=
a
x
where
a
is a constant.
a
x
( )
′
=
a
u
( )
′
=
Example 3
Using the Algebra of logarithms
At what point on the graph of the function
y
=
2
x

3 does the tangent line have slope 21?
Derivative of
y
=
log
a
x
Change to exponential form and differentiate implicitly.
y
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 Spring '10
 John
 Calculus, Derivative, Logarithmic Functions, Formulas, Natural logarithm, Logarithm, Calculus Section, new formulas

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