So far, we have learned how to differentiate a variety of functions, including trigonometric, inverse, and implicit functions. In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.

# Derivative of the Exponential Function

Just as when we found the derivatives of other functions, we can find the derivatives of exponential and logarithmic functions using formulas. As we develop these formulas, we need to make certain basic assumptions. The proofs that these assumptions hold are beyond the scope of this course.

First of all, we begin with the assumption that the function , where is a positive integer—as the product of multiplied by itself times. Later, we defined for a positive integer , and for positive integers and . These definitions leave open the question of the value of where is an arbitrary real number. By assuming the *continuity* of as where the values of as we take the limit are rational. For example, we may view as the number satisfying

As we see in the following table, .

64 | 77.8802710486 | ||

73.5166947198 | 77.8810268071 | ||

77.7084726013 | 77.9242251944 | ||

77.8162741237 | 78.7932424541 | ||

77.8702309526 | 84.4485062895 | ||

77.8799471543 | 256 |

We also assume that for of the derivative exists. In this section, we show that by making this one additional assumption, it is possible to prove that the function is differentiable everywhere.

We make one final assumption: that there is a unique value of for which . We define to be this unique value, as we did in Introduction to Functions and Graphs. (Figure) provides graphs of the functions , and . A visual estimate of the slopes of the tangent lines to these functions at 0 provides evidence that the value of lies somewhere between 2.7 and 2.8. The function is called the **natural exponential function**. Its inverse, is called the **natural logarithmic function**.

For a better estimate of , we may construct a table of estimates of for functions of the form . Before doing this, recall that

for values of very close to zero. For our estimates, we choose and to obtain the estimate.

See the following table.

2 | 2.7183 | ||

2.7 | 2.719 | ||

2.71 | 2.72 | ||

2.718 | 2.8 | ||

2.7182 | 3 |

The evidence from the table suggests that .

The graph of together with the line are shown in (Figure). This line is tangent to the graph of at .

Now that we have laid out our basic assumptions, we begin our investigation by exploring the derivative of exists. By applying the limit definition to the derivative we conclude that.

Turning to , we obtain the following.

We see that on the basis of the assumption that is differentiable at is not only differentiable everywhere, but its derivative is.

For . Thus, we have . (The value of for an arbitrary function of the form Let be the natural exponential function. Then.

In general,.

### Derivative of an Exponential Function

Find the derivative of .

#### Solution

Using the derivative formula and the chain rule,

### Combining Differentiation Rules

Find the derivative of .

#### Solution

Use the derivative of the natural exponential function, the quotient rule, and the chain rule.

Find the derivative of .

#### Solution

#### Hint

Don’t forget to use the product rule.

### Applying the Natural Exponential Function

A colony of mosquitoes has an initial population of 1000. After days, the population is given by . Show that the ratio of the rate of change of the population, , to the population size, is constant.

#### Solution

First find . By using the chain rule, we have . Thus, the ratio of the rate of change of the population to the population size is given by.

The ratio of the rate of change of the population to the population size is the constant 0.3.

If describes the mosquito population after days, as in the preceding example, what is the rate of change of after 4 days?

#### Solution

996

#### Hint

Find .

# Derivative of the Logarithmic Function

Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.

### The Derivative of the Natural Logarithmic Function

If , then.

More generally, let be a differentiable function. For all values of for which is given by.

## Proof

If , then . Differentiating both sides of this equation results in the equation.

Solving for yields.

Finally, we substitute to obtain.

We may also derive this result by applying the inverse function theorem, as follows. Since is the inverse of , by applying the inverse function theorem we have.

Using this result and applying the chain rule to yields

The graph of and its derivative are shown in (Figure).

### Taking a Derivative of a Natural Logarithm

Find the derivative of .

#### Solution

Use (Figure) directly.

### Using Properties of Logarithms in a Derivative

Find the derivative of .

#### Solution

At first glance, taking this derivative appears rather complicated. However, by using the properties of logarithms prior to finding the derivative, we can make the problem much simpler.

Differentiate: .

#### Solution

#### Hint

Use a property of logarithms to simplify before taking the derivative.

Now that we can differentiate the natural logarithmic function, we can use this result to find the derivatives of and for Let be a differentiable function.

- If , then.More generally, if , then for all values of for which .
- If , then.More generally, if , then.

## Proof

If , then . It follows that . Thus . Solving for , we have . Differentiating and keeping in mind that is a constant, we see that.

The derivative in (Figure) now follows from the chain rule.

If , then . Using implicit differentiation, again keeping in mind that is constant, it follows that Solving for and substituting , we see that.

The more general derivative ((Figure)) follows from the chain rule.

### Applying Derivative Formulas

Find the derivative of .

#### Solution

Use the quotient rule and (Figure).

### Finding the Slope of a Tangent Line

Find the slope of the line tangent to the graph of at .

#### Solution

To find the slope, we must evaluate at . Using (Figure), we see that.

By evaluating the derivative at , we see that the tangent line has slope.

Find the slope for the line tangent to at .

#### Solution

#### Hint

Evaluate the derivative at .

# Logarithmic Differentiation

At this point, we can take derivatives of functions of the form for certain values of , as well as functions of the form , where . Unfortunately, we still do not know the derivatives of functions such as or . These functions require a technique called** logarithmic differentiation**, which allows us to differentiate any function of the form . It can also be used to convert a very complex differentiation problem into a simpler one, such as finding the derivative of . We outline this technique in the following problem-solving strategy.

### Problem-Solving Strategy: Using Logarithmic Differentiation

- To differentiate using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain .
- Use properties of logarithms to expand as much as possible.
- Differentiate both sides of the equation. On the left we will have .
- Multiply both sides of the equation by to solve for .
- Replace by .

### Using Logarithmic Differentiation

Find the derivative of .

#### Solution

Use logarithmic differentiation to find this derivative.

### Using Logarithmic Differentiation

Find the derivative of .

#### Solution

This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.

### Extending the Power Rule

Find the derivative of where is an arbitrary real number.

#### Solution

The process is the same as in (Figure), though with fewer complications.

Use logarithmic differentiation to find the derivative of .

#### Solution

#### Hint

Follow the problem solving strategy.

Find the derivative of .

#### Solution

### Key Concepts

- On the basis of the assumption that the exponential function , and the relationship allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
- Logarithmic differentiation allows us to differentiate functions of the form or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.

# Key Equations

**Derivative of the natural exponential function****Derivative of the natural logarithmic function****Derivative of the general exponential function****Derivative of the general logarithmic function**