Problem 64 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln (6 x) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln(6x) \) is \( \frac{1}{x} \).

Step 1: Recall the Derivative Rule for Natural Logarithm

The derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). We will use this rule to find the derivative of \( \ln(6x) \).

Step 2: Identify the Inner Function

In the function \( f(x) = \ln(6x) \), the inner function \( u \) is \( 6x \). We need to differentiate \( u \) with respect to \( x \).

Step 3: Differentiate the Inner Function

Differentiate \( u = 6x \) with respect to \( x \). The derivative is \( \frac{du}{dx} = 6 \).

Step 4: Apply the Chain Rule

Apply the chain rule \( \frac{d}{dx} \ln(6x) = \frac{1}{6x} \cdot 6 \), by substituting \( u = 6x \) and \( \frac{du}{dx} = 6 \).

Step 5: Simplify

Simplify the expression:\[ \frac{1}{6x} \times 6 = \frac{6}{6x} = \frac{1}{x} \].

Key Concepts

Natural Logarithm

Understanding the concept of the natural logarithm is essential in calculus differentiation. The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It's widely used in mathematics because of its beautiful properties, especially when it comes to calculus. The natural logarithm has some important properties that make it unique:

• \(\ln(1) = 0\) because \( e^0 = 1 \).
• It is undefined for non-positive numbers; you can only take the natural log of positive numbers.
• \(\ln(e) = 1\) since \( e^1 = e \).

Natural logarithms simplify the process of differentiation, as they convert complex multiplication and division into addition and subtraction. This is why the \( \ln \) function is so useful in calculus.

Chain Rule

The chain rule is a fundamental tool in calculus used to find the derivative of composite functions. It lets us differentiate functions nestled within each other, like in the case of \( \ln(6x) \). When differentiating composite functions, you identify the 'outer' and 'inner' functions. The outer function in our example is \( \ln(u) \) with the inner function being \( u = 6x \). The chain rule formula for a composite function \( f(g(x)) \) is:

• Differentiate the outer function with respect to the inner function.
• Multiply this result by the derivative of the inner function.

Applying the chain rule allows us to break down complex differentiations into simpler steps. This way, even if the original function is complicated, we can handle it by tackling each part individually.

Derivative of ln Function

The derivative of the natural logarithm function has a specific rule that simplifies many differentiation tasks. The rule is that the derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \times \frac{du}{dx} \). For the function \( f(x) = \ln(6x) \), we apply this rule. The key steps are:

• Firstly, identify \( u \): Here, \( u = 6x \).
• Differentiating, \( \frac{du}{dx} = 6 \).
• Substitute into the derivative formula: \( \frac{1}{6x} \times 6 \).

By simplifying the expression, we find the final derivative is \( \frac{1}{x} \). This demonstrates how the natural logarithm’s derivative rule brings efficiency and accuracy to finding derivatives of functions involving natural logs.