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6.4

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6.4 Derivatives of Logarithmic

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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.

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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

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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

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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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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:

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Example 12

Using Formula 6 and the Chain Rule, we get

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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

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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.

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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

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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

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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 nas x → 0+ and so an alternative expression for e is

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