**6.4** Derivatives of Logarithmic

### Functions

2

### Derivatives of Logarithmic Functions

In this section we find the derivatives of the logarithmic
*functions y = log*_{b}*x and the exponential functions y = b** ^{x}*.

*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 = e** ^{x}*.

### Derivatives of Logarithmic Functions

In general, if we combine Formula 1 with the Chain Rule, we get

4

### 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*

6

### Example 9

Evaluate Solution:

*We make the substitution u = x*^{2} + 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
*x*^{2} *+ 1 > 0 for all x.*

We could use the properties of logarithms to write the answer as

but this isn’t necessary.

cont’d

8

### Derivatives of Logarithmic Functions

### General Logarithmic and

### Exponential Functions

10

### 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

12

### General Logarithmic and Exponential Functions

### Example 13

Combining Formula 7 with the Chain Rule, we have

14

### General Logarithmic and Exponential Functions

The integration formula that follows from Formula 7 is

### Logarithmic Differentiation

16

### 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(x*^{2} *+ 1) – 5 ln(3x + 2)*
*Differentiating implicitly with respect to x gives*

18

*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

20

*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

22

*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*

24

*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*