This article will focus on an intriguing task – finding the derivative of **ln****(2x)** (the n**atural logarithm function**). As one of the cornerstone concepts in **calculus**, the **derivative** serves as a powerful tool in deciphering the **rate of change** or the **slope** of a function at any point.

## Defining Derivative of ln(2x)

The **derivative** of a function measures how the function changes as its input changes. It’s often described as the function’s “**rate of change**” or the **slope** of the **tangent line** to the function’s graph at a specific point.

The derivative of **ln(2x),** written as **d/dx[ln(2x)]**, can be found by applying the **chain rule**, a basic theorem in **calculus**. The chain rule states that the derivative of a **composite function** is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

The derivative of the **natural logarithm function** **ln**(x) is **1/x**. And the derivative of **2x** with respect to **x** is **2**.

Figure-1.

Therefore, by the chain rule, the derivative of **ln(2x)** is:

d/dx[ln(2x)] = (1/(2x)) * 2

d/dx[ln(2x)] = 1/x

So, the derivative of **ln(2x)** is **1/x**.

**Properties of **Derivative of ln(2x)

The **derivative of ln(2x)** is **1/x**. This **derivative** has some key properties that are characteristic of **derivative functions** in general:

**Linearity**

The **derivative operator** is **linear**. This means that if you have two functions **u(x)** and **v(x)**, the derivative of their sum is the sum of their derivatives. However, as **ln(2x)** is a single function, this property is not explicitly reflected here.

**Local Information**

The **derivative** of a function at a particular point gives the **slope** of the **tangent line** to the graph of the function at that point. For the function **ln(2x)**, its derivative **1/x** is the slope of the tangent line to the graph of **ln(2x)** at any point **x**.

**Rate of Change**

The **derivative** of a function at a certain point gives the **rate of change** of the function at that point. For the function **ln(2x)**, its derivative **1/x** represents how fast ln(2x) is changing at any point **x**.

**Non-negativity for x > 0**

The **derivative** **1/x** is always positive for **x > 0**, which means that the **function ****ln(2x)** is increasing for **x > 0**. The greater the **x**, the slower the rate of increase (since **1/x** gets smaller as **x** gets larger).

**Undefined at x = 0**

The **derivative ****1/x** is undefined at **x = 0**, reflecting the fact that the function **ln(2x)** itself is undefined at **x = 0**.

**Negativity for x < 0**

The **derivative ****1/x** is always negative for **x < 0**, which means that the **function** **ln(2x)** is decreasing for **x < 0**. However, since the **natural logarithm** of a negative number is undefined in the **real number system**, this is typically not relevant in most **real-world applications**.

**Continuity and Differentiability**

The **derivative ****1/x** is **continuous** and **differentiable** for all **x ≠ 0**. This means that the function **ln(2x)** has a derivative at all such points, which informs us about the behavior and properties of the **original function**.

**Exercise **

**Example 1**

Compute **d/dx[ln(2x)]**

### Solution

The derivative of ln(2x) is 1/x.

**Example 2**

Determine **d/dx[2*ln(2x)]**

Figure-2.

### Solution

Here, we use the rule that the derivative of a constant times a function is the constant times the derivative of the function. So, the derivative is:

2*(1/x) = 2/x

**Example 3**

Compute **$d/dx[ln(2x)]^2$**

### Solution

We use the chain rule, which gives:

2*ln(2x)*(1/x) = 2ln(2x)/x

**Example 4**

Determine **d/dx[ln(2x + 1)]**

Figure-3.

### Solution

Here, the derivative is:

1/(2x + 1) * 2 = 2/(2x + 1)

**Example 5**

Compute **d/dx[ln(2x²)]**

### Solution

In this case, the derivative is:

1/(2x²) * 4x = 2/x

**Example 6**

Compute **d/dx[3ln(2x) – 2]**

Here, the derivative is:

3*(1/x) = 3/x

**Example 7**

Evaluate **d/dx[ln(2x) / x]**

Figure-4.

### Solution

Here we have a quotient, so we use the quotient rule for differentiation (d/dx [u/v] = (vu’ – uv’) / v²), where u = ln(2x) and v = x.

The derivative is then:

(x*(1/x) – ln(2x)*1) / x² = (1 – ln(2x)) / x

**Example 8**

Determine **d/dx[5ln(2x) + 3x²]**

### Solution

In this case, the derivative is:

5*(1/x) + 6x = 5/x + 6x

**Applications **

The derivative of ln(2x), which is 1/x, has broad applications across a variety of fields. Let’s explore some of these:

**Physics**

In physics, the concept of a **derivative** is fundamentally used to calculate **rates of change**. This concept finds wide application in various areas, such as **motion studies** where it helps determine **velocity** and **acceleration**. By taking derivatives of **displacement** with respect to **time**, we can obtain the **instantaneous velocity** and **acceleration** of an object.

**Economics**

In **economics**, the derivative of **ln(2x)** might be used in models where a **natural logarithm** is used to represent a **utility function** or **production function**. The derivative would then provide information about the **marginal utility** or **marginal product**.

**Biology**

In the study of population dynamics, the **natural logarithm** function often arises when examining **exponential growth** or **decay** (as in population growth or decay of biological specimens). The derivative, thus, helps in understanding the **rate of change** of the **population**.

**Engineering**

In **electrical engineering**, the **natural logarithm** and its derivative might be used in solving problems related to **signal processing** or **control systems**. Similarly, in **civil engineering**, it can be used in the analysis of **stress-strain behavior** of certain materials.

**Computer Science**

In **computer science**, particularly in **machine learning** and **optimization algorithms**, derivatives, including those of natural logarithms, are used to minimize or maximize **objective functions**, such as in **gradient descent**.

**Mathematics**

Of course, in **mathematics** itself, the derivative of **ln(2x)** and similar functions are frequently used in **calculus** in topics such as **curve sketching**, **optimization problems**, and **differential equations**.

*All images were created with GeoGebra.*