Problem 65 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln |5 x| $$

Step-by-Step Solution

Answer

\( f'(x) = \frac{1}{x} \).

Step 1: Understand the Function

We need to differentiate the function \( f(x) = \ln |5x| \). The absolute value inside the natural logarithm means we'll apply the chain rule during differentiation.

Step 2: Apply the Chain Rule

To differentiate \( \ln |5x| \), we apply the chain rule. This rule states that if \( u(x) \) is a function inside another function, then the derivative is \( \frac{d}{dx}\ln|u(x)| = \frac{1}{u(x)} \cdot u'(x) \). Here, \( u(x) = 5x \).

Step 3: Differentiate the Inner Function

First, find the derivative of \( u(x) = 5x \), which is \( u'(x) = 5 \).

Step 4: Differentiate the Logarithmic Function

Next, differentiate \( \ln |5x| \) using the result from Steps 2 and 3: \( \frac{d}{dx} \ln |5x| = \frac{1}{5x} \cdot 5 = \frac{5}{5x} = \frac{1}{x} \).

Step 5: Finalize the Differentiation

The value of the derivative does not depend on the sign of this function because \( \ln |5x| \) simplifies it for positive and negative values of \( x \). Thus, \( f'(x) = \frac{1}{x} \).

Key Concepts

Chain Rule

In calculus, the chain rule is a fundamental technique for differentiating composite functions. A composite function is essentially a function within another function, like nesting one box inside another.
To grasp the chain rule, imagine you have a function \( f(g(x)) \). To differentiate this, the chain rule tells us to differentiate the outer function, \( f \), while leaving the inner function, \( g(x) \), unchanged. Then, multiply this result by the derivative of the inner function, \( g(x) \). This can be written as:

• \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

With our exercise, we applied this rule to \( \ln |5x| \). We first treated the absolute value expression \( |5x| \) as the inner function, differentiating it independently. The chain rule is a versatile tool for functions with layered structures, making it indispensable in calculus.

Logarithmic Differentiation

Logarithmic differentiation is particularly handy for functions with multiplicative or exponential characteristics. This technique leverages the properties of logarithms to make differentiation simpler. When dealing with complex functions, logarithmic differentiation can transform them into a more approachable form.
For instance, given a function \( f(x) = \ln |5x| \), logarithmic differentiation allows us to differentiate the natural logarithm easily. The derivative of \( \ln u \) with respect to \( x \) is \( \frac{1}{u} \cdot u' \). Hence, to find the derivative of \( \ln |5x| \), we first consider the absolute term inside as \( u(x) = 5x \).

• The derivative becomes \( \frac{1}{5x} \) due to the logarithmic differentiation rule.

Logarithmic differentiation simplifies the process by breaking the function into manageable parts. This strategy is not just simple but also effective for a wide array of functions, underscoring the special role logarithms play in calculus.

Absolute Value Function

The absolute value function, denoted \( |x| \), is essential in understanding the behavior of functions over both positive and negative domains. It transforms a number into its non-negative form.
Consider a function like \( \ln |5x| \). The absolute value ensures the logarithm can handle both positive and negative inputs, as you typically cannot take the logarithm of a negative number.
With properties such as:

• \( |x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases} \)

The derivative of \( \ln |5x| \) becomes simpler. Regardless of whether \( x \) is positive or negative, \( |5x| \) equalizes the sign, making the function's behavior straightforward to interpret. This property highlights the significance of the absolute value in maintaining function validity across domains.