6.4 Derivatives of Logarithmic
Functions
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Derivatives of Logarithmic Functions
In this section we find the derivatives of the logarithmic functions y = logbx and the exponential functions y = bx. We start with the natural logarithmic function y = ln x.
We know that it is differentiable because it is the inverse of the differentiable function y = ex.
Derivatives of Logarithmic Functions
In general, if we combine Formula 1 with the Chain Rule, we get
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Example 2
Find ln(sin x).
Solution:
Using (2), we have
Derivatives of Logarithmic Functions
The corresponding integration formula is
Notice that this fills the gap in the rule for integrating power functions:
if n ≠ –1
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Example 9
Evaluate Solution:
We make the substitution u = x2 + 1 because the differential du = 2x dx occurs (except for the constant factor 2).
Thus x dx = du and
Example 9 – Solution
Notice that we removed the absolute value signs because x2 + 1 > 0 for all x.
We could use the properties of logarithms to write the answer as
but this isn’t necessary.
cont’d
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Derivatives of Logarithmic Functions
General Logarithmic and
Exponential Functions
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General Logarithmic and Exponential Functions
The logarithmic function with base b in terms of the natural logarithmic function:
Since ln b is a constant, we can differentiate as follows:
Example 12
Using Formula 6 and the Chain Rule, we get
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General Logarithmic and Exponential Functions
Example 13
Combining Formula 7 with the Chain Rule, we have
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General Logarithmic and Exponential Functions
The integration formula that follows from Formula 7 is
Logarithmic Differentiation
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Logarithmic Differentiation
The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms.
The method used in the next example is called logarithmic differentiation.
Example 15
Differentiate
Solution:
We take logarithms of both sides of the equation and use the properties of Logarithms to simplify:
ln y = ln x + ln(x2 + 1) – 5 ln(3x + 2) Differentiating implicitly with respect to x gives
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Example 15 – Solution
Solving for dy/dx, we get
Because we have an explicit expression for y, we can substitute and write
cont’d
Logarithmic Differentiation
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The Number e as a Limit
The Number e as a Limit
If f(x) = ln x, then f′(x) = 1/x. Thus f ′(1) = 1. We now use this fact to express the number e as a limit.
From the definition of a derivative as a limit, we have
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The Number e as a Limit
Because f′(1) = 1, we have
Then, by the continuity of the exponential function, we have
The Number e as a Limit
Formula 8 is illustrated by the graph of the function
y = (1 + x)1/x in Figure 7 and a table of values for small values of x. This illustrates the fact that, correct to seven decimal places,
e ≈ 2.7182818
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The Number e as a Limit
If we put n = 1/x in Formula 8, then n → as x → 0+ and so an alternative expression for e is
