Problem 66 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln |10 x| $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln |10x| \) is \( f'(x) = \frac{1}{x} \).

Step 1: Identify the Function to Differentiate

The given function is \( f(x) = \ln |10x| \). We need to find the derivative of this function with respect to \( x \).

Step 2: Apply the Chain Rule

Since \( f(x) = \ln |u| \) with \( u = 10x \), we use the chain rule to differentiate the function. The derivative of \( \ln |u| \) with respect to \( u \) is \( \frac{1}{u} \). So, \( \frac{d}{dx}(\ln |u|) = \frac{1}{u} \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Now, differentiate \( u = 10x \) with respect to \( x \). The derivative \( \frac{du}{dx} = 10 \).

Step 4: Combine the Derivatives

Substitute \( u = 10x \) and \( \frac{du}{dx} = 10 \) into the derivative expression from Step 2: \[ \frac{d}{dx}(\ln |10x|) = \frac{1}{10x} \times 10 = \frac{10}{10x} = \frac{1}{x} \].

Step 5: Final Result

The derivative of the function \( f(x) = \ln |10x| \) is \( f'(x) = \frac{1}{x} \).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It can be thought of as the rate of change or the slope of the function at a particular point. In notation, the derivative of a function \( f(x) \) with respect to \( x \) is represented as \( f'(x) \) or \( \frac{df}{dx} \). Calculating a derivative involves a few essential rules and techniques such as the power rule, product rule, quotient rule, and the chain rule.

The chain rule is particularly useful when dealing with composite functions, where one function is nested inside another. For example, when looking at \( f(x) = \ln |10x| \), we treat \( |10x| \) as an inner function. The chain rule states that the derivative of a composite function \( f(g(x)) \) is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This allows us to systematically break down more complex functions into simpler parts, calculating their derivatives piece by piece.

Logarithmic Differentiation

Logarithmic differentiation is especially useful when differentiating logarithms that are either messy or involve absolute values. It can make complex derivatives simpler to handle. When you differentiate a natural logarithm, such as \( \ln |u| \), you apply the rule \( \frac{d}{du}[\ln |u|] = \frac{1}{u} \). It’s crucial to remember to multiply by the derivative of \( u \) itself, which is derived using the chain rule.



Here’s how it works step-by-step:

• Identify the form: Recognize that \( f(x) = \ln |10x| \) includes an absolute value.
• Differentiate the outer function: The outer function, \( \ln |u| \), has a derivative of \( \frac{1}{u} \).
• Apply the chain rule: Multiply by the derivative of the inner function \( u = 10x \), which is just \( 10 \).
• Combine: Substitute back to achieve the final result, simplifying to \( \frac{1}{x} \).

Using logarithmic differentiation can simplify the differentiation process greatly, as it turns complicated multiplication or division into addition or subtraction.

Absolute Value Function

An absolute value function takes any real number input and returns its non-negative value. It is represented by \(|x|\), meaning it outputs \( x \) if \( x \geq 0 \) and \(-x\) if \( x < 0 \). When differentiating a function with an absolute value, we must consider the different cases caused by the absolute value.

The derivative of an absolute value function requires understanding its piecewise nature. However, when used within a natural logarithm as it is in this exercise \( \ln |10x| \), the absolute value affects the derivative calculation indirectly. You consider \( u = 10x \), ignoring initially whether \( x \) is positive or negative, because the derivative of \( \,|u|,\) specifically \( \ln |u| \), focuses on \( \frac{du}{dx} \), which smoothly resolves to its form by considering the sign change internally.

This systematic approach to differentiating functions involving absolute values, in conjunction with applying the chain rule, ensures that the derivative is accurately computed across all applicable domains of the original function.