Problem 2 - Calculus Early Transcendentals

Question

Find \(d y / d x\) $$ y=\ln \frac{x}{3} $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dx} \) is \( \frac{1}{x} \).

Step 1: Apply the Logarithmic Property

The given function is \(y = \ln \frac{x}{3}\), which can be rewritten using the property of logarithms: \(\ln \frac{x}{3} = \ln x - \ln 3\). This simplification helps in differentiating individual terms separately.

Step 2: Differentiate Each Term Separately

To find the derivative \( \frac{dy}{dx} \), differentiate each term from the expression in Step 1. The derivative of \( \ln x \) with respect to \(x\) is \( \frac{1}{x} \), and the derivative of the constant \( \ln 3 \) is zero.

Step 3: Combine the Derivatives

Combine the derivatives obtained from Step 2. Thus, the derivative \( \frac{dy}{dx} \) is \( \frac{1}{x} - 0 = \frac{1}{x} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used when differentiating complex functions, particularly products, quotients, or powers. The core idea is to apply logarithms to both sides of an equation, transforming the complexity into simpler parts that are easier to differentiate. This method simplifies multiplication into addition, division into subtraction, and powers into products, using properties of logarithms. Once the function is expressed in terms of its logarithm, derivatives are easier to handle. To start, take the natural logarithm of both sides of the function you want to differentiate. Then, differentiate implicitly with respect to the variable, following the rules of differentiation. This helps when direct differentiation is tricky or when dealing with functions like products or quotients. Logarithmic differentiation is especially useful for simplifying derivatives of functions that involve chains or need the quotient rule repeatedly.

Properties of Logarithms

Understanding the properties of logarithms is key to simplifying functions before differentiating them. These properties allow us to break down functions into manageable parts that are easier to differentiate. Here are some fundamental properties:

• Product Rule: \( \ln(ab) = \ln a + \ln b \). This property helps split products into sums.
• Quotient Rule: \( \ln \frac{a}{b} = \ln a - \ln b \). This is especially useful for transforming division into subtraction.
• Power Rule: \( \ln(a^b) = b \ln a \). This allows powers to be multiplied out front, making differentiation easier.

For example, in our exercise: \( \ln \frac{x}{3} \) becomes \( \ln x - \ln 3 \). By applying the quotient rule, we separate the term into a difference, making it straightforward to differentiate each part separately. This ability to decompose functions using logarithm properties is powerful for derivative calculations.

Natural Logarithm Differentiation

Differentiating natural logarithms involves understanding the derivative of the natural log function, \(\ln x\), with respect to \(x\). The derivative of \(\ln x\) is \(\frac{1}{x}\). This foundational concept simplifies derivative computations whenever \(\ln x\) appears in functions. In the exercise provided, by transforming \(y = \ln \frac{x}{3}\) into \(\ln x - \ln 3\), each log term can be differentiated. The differentiation results in:

• The derivative of \(\ln x\) is \(\frac{1}{x}\).
• The derivative of \(\ln 3\) is zero because it is a constant.

Thus, the function's complete derivative becomes \(\frac{1}{x} - 0 = \frac{1}{x}\). This illustrates how efficiently we can handle natural logarithm functions and find derivatives using the rules of differentiation combined with the properties of logarithms.