Problem 60 - Calculus and its applications

Question

Differentiate. $$ y=4 \ln |x| $$

Step-by-Step Solution

Answer

The derivative of \( y = 4 \ln |x| \) is \( y' = \frac{4}{x} \).

Step 1: Identify the function

The function given is \( y = 4 \ln |x| \). This is a logarithmic function where \( y \) is defined in terms of the natural logarithm of the absolute value of \( x \). The constant multiplier is 4.

Step 2: Recall the derivative rule for ln

The derivative of \( \ln|x| \) with respect to \( x \) is \( \frac{1}{x} \). The derivative of a constant times a function is the constant times the derivative of the function.

Step 3: Apply the derivative rule

Taking the derivative of \( y = 4 \ln |x| \), we apply the constant multiplication rule and the logarithmic derivative rule: \( y' = 4 \cdot \frac{d}{dx}(\ln|x|) = 4 \cdot \frac{1}{x} \).

Step 4: Simplify the expression

Simplify the expression obtained from the previous step: \( y' = \frac{4}{x} \). This is the derivative of the given function with respect to \( x \).

Key Concepts

Understanding Logarithmic Differentiation

Logarithmic differentiation is an invaluable technique in calculus used to find the derivative of functions that are not easily handled by standard differentiation rules. It's especially useful when dealing with products, quotients, or powers of functions. In our exercise, however, it simplifies the differentiation process by leveraging the properties of logarithms and derivatives.
Use logarithmic differentiation in the following scenarios:

• When differentiating products of many functions
• If you have functions with variables in both the base and the exponent
• If separating complex expressions makes them simpler to differentiate

To implement this technique, you first take the natural logarithm of both sides of the equation, then differentiate implicitly with respect to the variable of interest. This allows taking advantage of properties like turning exponents into coefficients. It ultimately simplifies the entire differentiation process.

Exploring the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), an irrational constant approximately equal to 2.71828. The natural logarithm is widely utilized in calculus because it naturally arises in the analysis of growth processes such as exponential growth and compound interest.
The function \( \ln|x| \) is crucial because, unlike \( \ln(x) \), it considers the domain of all real numbers except zero while handling absolute values. This means it can handle negative inputs by taking the absolute value of \( x \) before the logarithmic operation.
Here are key points about natural logarithms:

• The derivative of \( \ln|x| \) is \( \frac{1}{x} \), as seen in our original exercise.
• The natural logarithm is only defined for positive values of inputs in usual contexts, but \( \ln|x| \) extends this by incorporating absolute values.
• \( \ln(e) = 1 \), providing a foundation for simplifying expressions with \( \ln(x) \).

These properties make \( \ln(x) \) an essential tool in calculus for solving logarithmic differentiation and other related problems.

Foundations of Calculus and Differentiation

Calculus is the mathematical study of continuous change and forms the foundation for many scientific disciplines. Differentiation, one of its core components, is the process of finding the rate at which a function changes at any given point. This concept is crucial for analyzing and solving real-world problems involving dynamic systems.
The derivative is symbolically represented as \( f'(x) \) or \( \frac{dy}{dx} \) and gives the slope of the tangent line to the curve of the function at any point.
In our specific example, differentiating the logarithmic function involves applying simple rules:

• The derivative of \( \ln|x| \) is \( \frac{1}{x} \).
• Multiplying this derivative by the constant, 4, gives us \( \frac{4}{x} \), which shows how fast the function \( y = 4 \ln |x| \) changes with respect to \( x \).

Differentiation helps in understanding how variables affect each other and is fundamental in optimization, curve sketching, and solving differential equations. Understanding how to differentiate functions, particularly those involving logarithmic expressions, opens the door to deeper mathematical analysis and problem-solving.