Problem 11 - Thomas Calculus

Question

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \frac{3}{x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

We are given the function \( y = \ln \frac{3}{x} \). Our goal is to find \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \).

Step 2: Use Logarithm Property

Apply the property of logarithms that states \( \ln \frac{a}{b} = \ln a - \ln b \). Rewriting the function gives: \[ y = \ln 3 - \ln x \].

Step 3: Differentiate Each Term

Differentiate each term separately. The derivative of a constant, \( \ln 3 \), is 0. The derivative of \( -\ln x \) is \( -\frac{1}{x} \).

Step 4: Combine the Derivatives

Combine the derivatives from the previous step to find:\[ \frac{dy}{dx} = 0 - \frac{1}{x} = -\frac{1}{x} \].

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are in the form of a quotient, product, or powers, particularly when these forms make direct differentiation cumbersome. By taking the natural logarithm of both sides of an equation, complex expressions can be simplified using logarithmic properties. This method is very useful when dealing with functions where variables are present in both the base and the exponent.

• It simplifies the differentiation by transforming products into sums, quotients into differences, and powers into simple multiplications.
• Ideal for differentiating expressions like exponential functions where both the base and the exponent have variables.

It is helpful because taking the derivative directly might be very complicated, whereas logging the function reduces this complexity significantly. For instance, when differentiating a function like \( y = x^x \), using logarithmic differentiation can turn a multiplication and exponentiation problem into a more manageable form.

Logarithm Properties

The properties of logarithms are essential tools that simplify the process of differentiation by allowing us to break complex expressions into simpler terms. The most common property used is that the logarithm of a quotient is the difference of the logarithms: \[ \ln \left( \frac{a}{b} \right) = \ln a - \ln b \]In the given exercise, this property helps break down the complex function \( y = \ln \frac{3}{x} \) into simpler terms, \( y = \ln 3 - \ln x \). This is beneficial because:

• It transforms a division into a subtraction, which is simpler to differentiate.
• Allows constant terms, like \( \ln 3 \), to become zero easily during differentiation.

Understanding these properties assists in managing and simplifying logarithmic expressions, making it easier to identify derivative operations and ultimately solve the exercise effectively.

Chain Rule

The Chain Rule in calculus is a vital method for differentiating composite functions, which are functions made up by combining two or more functions. Essentially, it states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Mathematically, this is expressed as:\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]In our context, although logarithmic differentiation was mainly applied without explicitly using the Chain Rule, understanding it helps in cases where the argument of a logarithm is itself a complex function. For example:

• When differentiating \( \ln(x^2) \), you would apply the Chain Rule to find: \( \frac{d}{dx}[\ln(u)] \cdot \frac{d}{dx}[x^2] \).
• This results in \( \frac{1}{u} \cdot 2x \), simplifying to \( \frac{2x}{x^2} \).

The Chain Rule ensures accuracy when differentiating nested or hierarchically structured functions by methodically approaching each layer of the function. This guarantees that no component is left incorrectly differentiated, which could lead to errors in calculus problems.