Problem 80 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln \left(\frac{x^{2}+5}{x}\right) $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = \frac{2x}{x^2 + 5} - \frac{1}{x} \).

Step 1: Apply the Logarithm Rule

First, let's use the properties of logarithms to simplify the function: \[ f(x) = \ln \left( \frac{x^2 + 5}{x} \right) = \ln(x^2 + 5) - \ln(x) \] This applies the rule \(\ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b)\).

Step 2: Differentiate Each Term

Differentiate each term separately. For \(\ln(x^2 + 5)\), use the chain rule: \[ \frac{d}{dx} \ln(x^2 + 5) = \frac{1}{x^2 + 5} \cdot \frac{d}{dx}(x^2 + 5) = \frac{2x}{x^2 + 5} \] For \(\ln(x)\), \[ \frac{d}{dx} \ln(x) = \frac{1}{x} \]

Step 3: Combine the Derivatives

Now combine the derivatives from Step 2: \[ f'(x) = \frac{2x}{x^2 + 5} - \frac{1}{x} \] This expression represents the derivative of the original function.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique in calculus used to simplify the differentiation process, especially when dealing with complex functions. In this particular exercise, applying logarithmic differentiation made it easier to work with the original function.

• We started with the function \( f(x)=\ln \left(\frac{x^{2}+5}{x}\right) \).
• By using logarithmic properties, specifically \(\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)\), we simplified the function to \( \ln(x^2 + 5) - \ln(x) \).

This step was crucial as it broke down a complex expression into simpler parts. Another advantage of logarithmic differentiation is that it handles divisions and products within a function with ease, transforming them into more straightforward additions and subtractions of simpler functions.
This simplicity makes further steps, like applying the chain rule, more approachable.

Chain Rule

The chain rule is an essential differentiation technique used when differentiating composite functions. For a function given by \( f(g(x)) \), the chain rule states:

• The derivative is \( f'(g(x)) \, g'(x) \), where \( f'(g(x)) \) refers to the derivative of the outer function and \( g'(x) \) is the derivative of the inner function.

In our exercise, when we differentiated \( \ln(x^2 + 5) \), we experienced a classic application of the chain rule.

• Although the function looks simple, it's a composite: the outer function \( \ln(u) \) with \( u = x^2 + 5 \).
• The derivative of \( \ln(u) \) is \( \frac{1}{u} \), applied as \( \frac{1}{x^2 + 5} \).
• The derivative of \( x^2 + 5 \) is \( 2x \).

By linking these derivatives together with multiplication, the chain rule made differentiation possible and straightforward, resulting in \( \frac{2x}{x^2 + 5} \). The rule shines particularly in layered functions like these.

Calculus Problem Solving

Solving calculus problems involves understanding and accurately applying rules and techniques, like logarithmic differentiation and the chain rule, to find derivatives. Each step in problem solving is like a building block leading to the final derivative of a function.
Here’s how the process generally unfolds:

• Identify the Function: First, understand the function you need to differentiate.
• Apply Simplification Techniques: Use strategies like logarithmic properties to break the function into simpler parts.
• Differentiation: Use rules such as the chain rule and power rule to find the derivatives of the simplified parts.
• Combine Results: Piece together the derivatives you’ve found to obtain the final solution.

For the function \( f(x)= \ln \left(\frac{x^{2}+5}{x}\right) \), we effectively used these strategies by first simplifying the logarithms, then executing the differentiation. Finally, combining each derivative part arrived at the derivative \( f'(x) = \frac{2x}{x^2 + 5} - \frac{1}{x} \).
This methodical approach to problem solving ensures that errors are minimized and complex problems become manageable.