Problem 70 - Calculus Early Transcendentals

Question

Derivative of In \(|x|\) Differentiate \(\ln x\) for \(x>0\) and differentiate \(\ln (-x)\) for \(x<0\) to conclude that \(\frac{d}{d x}(\ln |x|)=\frac{1}{x}\).

Step-by-Step Solution

Answer

Answer: The derivative of the function \(y = \ln|x|\) is \(y' = \frac{1}{x}\)

Step 1: Differentiating \(\ln x\) for \(x>0\)

First, we will find the derivative of \(\ln x\) for positive \(x\) values. By definition, \(\frac{d}{d x}\ln x=\frac{1}{x}\) when \(x>0\).

Step 2: Differentiating \(\ln(-x)\) for \(x<0\)

Next, we will find the derivative of \(\ln(-x)\) for negative \(x\) values. Using chain rule, \(\frac{d}{d x}\ln(-x)=\frac{1}{-x}\cdot\frac{d(-x)}{d x}=\frac{1}{-x}\cdot(-1)=\frac{1}{x}\) when \(x<0\).

Step 3: Combining the results

Having found the derivatives of both parts of the function, we can now conclude that \(\frac{d}{d x}(\ln|x|)=\frac{1}{x}\) holds for both positive and negative \(x\) values, as both parts yield the same result in their respective domains.

Key Concepts

Absolute Value

The absolute value of a number is a measure of its magnitude without regard to its sign. It is denoted as \(|x|\) and turns any number into a positive value or zero. So, \(|x|\) equals \(x\) if \((x > 0)\) and \(-x\) if \((x < 0)\).
In the context of differentiation, \( ext{ln}(|x|)\) involves the absolute value function. This means we handle the \( ext{ln}\) function separately for positive and negative inputs to ensure the argument is a valid positive number. That's why we consider \( ext{ln}(x)\) when \((x > 0)\) and \( ext{ln}(-x)\) when \((x < 0)\), leading us to the unified result of \(\text{ln}(|x|)\).
The handling of the absolute value enables a smooth transition between positive and negative domains, ensuring that the derivative applies consistently across the entire real number line.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is approximately 2.718. It's a fundamental function in calculus due to its properties in growth and rate change.
When differentiating the natural logarithm, one useful property is that the derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). This is only valid for positive values of \(x\), since logarithms of non-positive numbers are undefined in the real number system. This base formula is key in various differentiation problems involving logarithms.
In problems requiring the natural logarithm differentiation across a domain with negative potential, such as \(\ln(|x|)\), the function must be adjusted so that the input remains positive. Exploring \(\ln(x)\) for \(x>0\) and \(\ln(-x)\) for \(x<0\) allows the use of the aforementioned derivative rule in their respective domains.

Chain Rule

The chain rule is a fundamental differentiation rule used when dealing with composite functions. It provides a method to differentiate a function that is the result of one function applied to another function. You can think of it as peeling back the layers of an onion for differentiation.
In this scenario, the chain rule is particularly useful when differentiating \(\ln(-x)\) for \(x < 0\). The chain rule states that if you have a function \(y = f(g(x))\), then the derivative \(y'\) is \((f'(g(x))) imes g'(x)\).

• For \(\ln(-x)\), treat \(-x\) as the inner function.
• The derivative of \(\ln(g(x))\) is \(\frac{1}{g(x)}\).
• Then multiply by \(\frac{d}{dx}(-x)\), which is \-1\.

the chain rule helps integrate the steps needed to properly apply differentiation to nested functions like these, enabling accurate derivatives over diverse scenarios.

Derivative Rules

Derivative rules are the formulas and techniques used to find derivatives of different functions with ease. Familiarity with these rules makes complex differentiation problems much simpler.
Some fundamental derivative rules include:

• The Power Rule: \(\frac{d}{dx}x^n = nx^{n-1}\)
• The Product Rule: \(\frac{d}{dx}(uv) = u'v + uv'\)
• The Quotient Rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\)

Another key rule is the derivative of the natural logarithm, which states \(\frac{d}{dx}\ln(x) = \frac{1}{x}\) for \(x > 0\). This simplifies calculations involving logarithmic functions.
Together, these rules make differentiation intuitive and manageable, offering straightforward solutions to seemingly complex calculus problems. By applying the most appropriate rule in the right scenarios, finding the derivative of functions like \(\ln(|x|)\) becomes a systematic and less daunting task.