# Differentiation Of The Natural Log Function Homework Solutions

Related Topics: More 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)}=xln(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.

**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.

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