Problem 29 - Essential Calculus Early Transcendentals

Question

Differentiate the function. $$ y=\ln \left|2-x-5 x^{2}\right| $$

Step-by-Step Solution

Answer

The derivative is \( \frac{-1 - 10x}{2 - x - 5x^2} \).

Step 1: Identify the Function Type

The function given is a natural logarithm of an absolute value: \( y = \ln |2 - x - 5x^2| \). To differentiate it, we'll use the chain rule and the derivative of a natural logarithm function.

Step 2: Apply the Derivative of ln Function

Recall that the derivative of \( \ln |u| \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = 2 - x - 5x^2 \).

Step 3: Differentiate the Inner Function

Find \( \frac{du}{dx} \) where \( u = 2 - x - 5x^2 \). The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = -1 - 10x \).

Step 4: Combine Results Using Chain Rule

Substitute back into the chain rule formula: \( \frac{dy}{dx} = \frac{1}{2 - x - 5x^2} \cdot (-1 - 10x) \). Simplifying, we get \( \frac{dy}{dx} = \frac{-1 - 10x}{2 - x - 5x^2} \).

Key Concepts

natural logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \) is approximately equal to 2.71828. It is one of the most important functions in mathematics, especially in calculus and complex analysis. The natural logarithm of a number provides the power to which \( e \) must be raised to produce that number.

In calculus, the natural logarithm function exhibits unique properties that make differentiation relatively straightforward. Specifically, the derivative of \( \ln |u| \) can be found using the formula \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u \) is a differentiable function of \( x \).

This formula highlights the application of the chain rule—a fundamental principle when working with composite functions that includes logarithms. It effectively allows us to peel apart the complex structure of composite functions into simpler, derivable components.

differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In calculus, it is one of the core operations used to uncover the behaviors of mathematical models and functions.

Here’s a brief review of some key concepts involved in differentiation:

• Derivatives: The derivative of a function \( y = f(x) \) is denoted \( \frac{dy}{dx} \) or \( f'(x) \), and it describes how \( y \) changes with respect to changes in \( x \).
• Chain Rule: This rule is indispensable when dealing with composite functions. It provides a method to differentiate a function dependent on another differentiable function. The chain rule states that if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).
• Power Rule: For derivatives, if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). This rule can simplify many expressions when combined with the chain rule.

These principles were essential in differentiating the function \( y = \ln |2 - x - 5x^2| \), using the derivative of \( \ln |u| \) and the chain rule to simplify the task.

absolute value

The absolute value refers to the non-negative value of any real number, ignoring the sign. It is denoted \( |x| \) and represents the distance of \( x \) from zero on the number line.

Mathematically, the absolute value function is defined as:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

When solving calculus problems, such as differentiation with natural logarithms, handling the absolute value becomes crucial because the expression inside the absolute value, called the argument, could potentially change signs. The derivative of \( \ln |x| \) is slightly different from the derivative of \( \ln x \), because it accounts for both positive and negative values of \( x \).

This specificity is important because it ensures that the function behaves correctly across all real numbers, which is particularly useful in applications of calculus and beyond. In the given exercise, \( \ln |2-x-5x^2| \) required special attention as the absolute value impacted how the derivative was constructed and simplified.