Problem 79 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln \left(\frac{x^{2}-7}{x}\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-x^3 + 2x^2 + 7x}{x(x^2 - 7)} \).

Step 1: Identify the Type of Function

The given function is a logarithmic function, specifically a natural log, which is \( f(x) = \ln\left(\frac{x^2 - 7}{x}\right) \). The derivative of a natural log function \( \ln(u) \) is given by \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u \) is the argument of the logarithm.

Step 2: Apply Logarithmic Properties

Use the property \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) to rewrite the function: \( f(x) = \ln(x^2 - 7) - \ln(x) \). This step simplifies the differentiation process.

Step 3: Differentiate Each Part Separately

Differentiate \( \ln(x^2 - 7) \) and \( \ln(x) \) separately. The derivative of \( \ln(x^2 - 7) \) is \( \frac{1}{x^2 - 7} \cdot (2x) \) using the chain rule, and the derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Step 4: Apply the Derivatives

Substitute the derivatives back into the expression: \( f'(x) = \frac{2x}{x^2 - 7} - \frac{1}{x} \).

Step 5: Simplify the Expression

To simplify, find a common denominator, which is \( x(x^2 - 7) \), giving: \( f'(x) = \frac{2x^2 - x(x^2 - 7)}{x(x^2 - 7)} \). This simplifies further to \( f'(x) = \frac{2x^2 - x^3 + 7x}{x(x^2 - 7)} \). Simplifying gives: \( f'(x) = \frac{-x^3 + 2x^2 + 7x}{x(x^2 - 7)} \).

Key Concepts

Natural Logarithm

Natural logarithms use the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. Natural logarithms are denoted by \( \ln(x) \). They are important in many areas of mathematics and are especially useful in calculus and exponential growth and decay models.

Natural logarithms have several key properties:

• The natural log of 1 is 0, i.e., \( \ln(1) = 0 \).
• The natural log of \( e \) is 1, i.e., \( \ln(e) = 1 \).
• Natural logs convert multiplication into addition, e.g., \( \ln(ab) = \ln(a) + \ln(b) \).
• They convert division into subtraction, as in \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

In the differentiation exercise, the natural logarithm \( \ln\left(\frac{x^2 - 7}{x}\right) \) was used. By applying the property that converts division into subtraction, it became easier to differentiate each component separately.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. Whenever you have a function within a function, you apply the chain rule to find the derivative most effectively. It states that the derivative of a composite function \( f(g(x)) \) is the derivative of \( f \) with respect to \( g \) multiplied by the derivative of \( g \) with respect to \( x \). In mathematical terms, this is expressed as:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In the given exercise, the chain rule is used in the differentiation of \( \ln(x^2 - 7) \). Here, the inside function \( x^2 - 7 \) differentiates to \( 2x \), which when multiplied by the outer function's derivative \( \frac{1}{x^2 - 7} \), gives the result of \( \frac{2x}{x^2 - 7} \).

This shows the effectiveness of the chain rule in handling complex expressions.

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique to differentiate functions that are products or quotients of other functions or have variables in the base and exponent. It employs logarithms to transform the original function into a form that is simpler to differentiate. This method is particularly helpful when dealing with complicated products or when using properties of logarithms to split functions into manageable parts.

The steps generally involve:

• Taking the natural logarithm of both sides of the function \( y = f(x) \), resulting in \( \ln(y) = \ln(f(x)) \).
• Applying the chain rule to differentiate \( \ln(y) \) with respect to \( x \), which results in \( \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}[\ln(f(x))] \).
• Finally, solving for \( \frac{dy}{dx} \) by multiplying both sides by \( y \) (or \( f(x) \)).

In the original problem, logarithmic differentiation simplifies the process by splitting \( \ln\left(\frac{x^2 - 7}{x}\right) \) into separate parts. This eliminates the complexity of differentiation through successive application of the chain rule and ensures a more straightforward solution path.