3.6 Derivatives of Logarithmic Functions
Marius Ionescu
10/06/2010
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
The derivative of log
Fact
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
The derivative of log
Fact
d
1
(ln x ) =
dx
x
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
The derivative of log
Fact
d
1
(ln x ) =
dx
x
d
1
(loga x ) =
dx
x ln a
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
The chain rule
Fact
d
g 0 (x )
ln g (x ) =
.
dx
g (x )
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
Dierentiate
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3.6 Derivatives of Logarithmic Functions
Examples
Example
Dierentiate
y = ln(cos x )
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
Dierentiate
y = ln(cos x )
√
y = 3 ln x.
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
Dierentiate
y = ln(cos x )
√
y = 3 ln x.
y = x ln(1 + e x )
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
Dierentiate
y = ln(cos x )
√
y = 3 ln x.
y = x ln(1 + e x )
+ln x
y = 11−
ln x
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Example
Example
Dierentiate
y = ln |x |.
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Example
Example
Dierentiate
y = ln |x |.
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Logarithmic Dierentiation
Fact (Steps in Logarithmic dierentiation)
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Logarithmic Dierentiation
Fact (Steps in Logarithmic dierentiation)
Take natural logarithms of both sides of an equation y = f (x )
and use the Laws of Logarithm to simplify
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Logarithmic Dierentiation
Fact (Steps in Logarithmic dierentiation)
Take natural logarithms of both sides of an equation y = f (x )
and use the Laws of Logarithm to simplify
Dierentiate implicitly with respect to x
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Logarithmic Dierentiation
Fact (Steps in Logarithmic dierentiation)
Take natural logarithms of both sides of an equation y = f (x )
and use the Laws of Logarithm to simplify
Dierentiate implicitly with respect to x
Solve the resulting equation for y 0 .
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
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3.6 Derivatives of Logarithmic Functions
Examples
Example
√
2
y = xe x (x 2 + 1)10
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
√
2
y = xe x (x 2 + 1)10
y = xx
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
√
2
y = xe x (x 2 + 1)10
y = xx
y = x cos x
Marius Ionescu
3.6 Derivatives of Logarithmic Functions
Examples
Example
y
y
y
y
=
√
2
xe x (x 2 + 1)10
= xx
= x cos x
= (ln x )sin x
Marius Ionescu
3.6 Derivatives of Logarithmic Functions