In these lessons, we will learn how to find the derivative of the natural log function (ln).

Related Topics: More Calculus Lessons

### Natural Log (Ln)

### Derivative of ln(x)

**Derivatives of Logarithmic Functions**

The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

**What are the formulas for finding derivatives of logarithmic functions and how to use them to find derivatives?**

The following are the formulas for the derivatives of logarithmic functions:

Examples:

Find the derivatives for the following logarithmic functions:

1. f(x) = ln(x^{2} + 10)

2. f(x) = √x ˙ ln(x)

3. f(x) = ln[(2x + 1)^{3}/(3x - 1)^{4}]

4. y = [log_{a}(1 + e^{x})]^{2}
**Derivatives of Logarithmic Functions**

Find the derivatives for the following logarithmic functions:

Examples:

1. y = ln(x^{2} x)

2. y = (log_{7} x)^{1/3}

3. y = ln(x^{4}˙sin x)

4. y = lnx/[1 + ln(2x)]

**Derivatives of the Natural Log Function (Basic)**

How to differentiate the natural logarithmic function?

Examples:

Determine the derivative of the function.

1. f(x) = 2ln(x)

2. f(x) = ln(4x)**Derivatives of the Natural Log Function with the Chain Rule**

How to differentiate the natural logarithmic function using the chain rule?

Example:

Determine the derivative of the function.

f(x) = 5ln(x^{3})
**The Derivative of the Natural Log Function**

We give two justifications for the formula for the derivative of the natural log function. If you want to see where this formula comes from, this is the video to watch.

Related Topics: More Calculus Lessons

The Natural Log is the logarithm to the base e. where e is an irrational constant approximately equal to 2.718281828. The natural logarithm is usually written ln(x) or log_{e}(x).

The natural log is the inverse function of the exponential function. They are related by the following identities:

e^{ln(x)} = x

ln(e^{x}) = x

Using the Chain Rule, we get

**Example:**

Differentiate y = ln(x^{2} +1)

**Solution:**

Using the Chain Rule, we get

**Example:**

Differentiate

**Solution:**

The derivative of the natural logarithmic function (ln[x]) is simply 1 divided by x. This derivative can be found using both the definition of the derivative and a calculator. Derivatives of logarithmic functions are simpler than they would seem to be, even though the functions themselves come from an important limit in Calculus.

The following are the formulas for the derivatives of logarithmic functions:

Examples:

Find the derivatives for the following logarithmic functions:

1. f(x) = ln(x

2. f(x) = √x ˙ ln(x)

3. f(x) = ln[(2x + 1)

4. y = [log

Find the derivatives for the following logarithmic functions:

Examples:

1. y = ln(x

2. y = (log

3. y = ln(x

4. y = lnx/[1 + ln(2x)]

How to differentiate the natural logarithmic function?

Examples:

Determine the derivative of the function.

1. f(x) = 2ln(x)

2. f(x) = ln(4x)

How to differentiate the natural logarithmic function using the chain rule?

Example:

Determine the derivative of the function.

f(x) = 5ln(x

We give two justifications for the formula for the derivative of the natural log function. If you want to see where this formula comes from, this is the video to watch.

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