**Related Pages**

Natural Logarithm

Logarithmic Functions

Derivative Rules

Calculus Lessons

### Natural Log (ln)

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

### Derivative Of ln(x)

Using the Chain Rule, we get

**Example:**

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

**Solution:**

Using the Chain Rule, we get

**Example:**

Differentiate

**Solution:**

### 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.

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

**Examples:**

Find the derivatives for the following logarithmic functions:

- f(x) = ln(x
^{2} + 10)
- f(x) = √x ˙ ln(x)
- f(x) = ln[(2x + 1)
^{3}/(3x - 1)^{4}]
- y = [log
_{a}(1 + e^{x})]^{2}

#### Derivatives Of Logarithmic Functions

Find the derivatives for the following logarithmic functions:

**Examples:**

- y = ln(x
^{2} x)
- y = (log
_{7} x)^{1/3}
- y = ln(x
^{4}˙sin x)
- 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.

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

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