Problem 2 - Calculus - AP Edition

Question

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

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{1}{x} \)

Step 1: Apply the Chain Rule

To find the derivative of the function \(y = \ln \frac{x}{3}\),we begin by recognizing it as a logarithmic function. We can rewrite the function using properties of logarithms: \[y = \ln x - \ln 3\]This lets us separate the terms. We will differentiate each term separately.

Step 2: Differentiate Each Term

Now differentiate the function:\( \frac{d}{dx}(\ln x - \ln 3) \). Since \( \ln 3 \) is a constant, its derivative is 0.The derivative of \( \ln x \) is \( \frac{1}{x} \), so:\[ \frac{d}{dx}(\ln x - \ln 3) = \frac{1}{x} - 0 = \frac{1}{x} \]

Step 3: Conclude the Derivative

The derivative of the function \( y = \ln \frac{x}{3} \) has been simplified to:\[ \frac{dy}{dx} = \frac{1}{x} \]This concludes the differentiation process.

Key Concepts

Chain Rule

The chain rule is an essential method used in calculus for finding the derivative of composite functions. It helps us to break down complex functions into simpler ones. This process is particularly useful when dealing with functions within functions.

• The chain rule states that if we have two functions, say \(f(x)\) and \(g(x)\), then the derivative of their composition \(f(g(x))\) is given by \(f'(g(x)) \cdot g'(x)\).
• This rule allows us to differentiate each function separately and then combine their derivatives.

In the context of our problem, we use the chain rule to identify the inner and outer functions when necessary. However, after applying properties of logarithms, we simplified the expression, which eliminated the need to explicitly use the chain rule.

Logarithmic Differentiation

Logarithmic differentiation is a technique that brings ease to differentiating functions that involve logarithms. It simplifies expressions before differentiation, making complex derivatives manageable.

Here's how it works:

• First, use the properties of logarithms to rewrite the function in a simpler form.
• Differentiate the transformed function term by term.

In our exercise, the original function was \( y = \ln \frac{x}{3} \). Recognizing the property of logarithms, we rewrote it as \( y = \ln x - \ln 3 \) to facilitate easier differentiation.
This separation simplifies the problem, as the derivative of a constant \(\ln 3 \) is zero, leaving us with only the derivative of \( \ln x \).

Properties of Logarithms

Properties of logarithms are powerful tools used in simplifying expressions that involve logarithms. These properties help transform complex logarithmic expressions into easier forms for computation and differentiation.

Key properties include:

• The quotient rule: \(\ln \left( \frac{a}{b} \right) = \ln a - \ln b\).
• The product rule: \(\ln (ab) = \ln a + \ln b\).
• The power rule: \(\ln (a^b) = b \cdot \ln a\).

In the original exercise, we applied the quotient rule to rewrite \( y = \ln \frac{x}{3} \) as \( y = \ln x - \ln 3 \). This simplification allowed for straightforward differentiation, highlighting how essential these logarithm properties are.