Problem 34 - Calculus: Early Transcendentals

Question

(a) If \( f(x) = x + 1/x \), find \( f'(x) \). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of \( f \) and \( f' \).

Step-by-Step Solution

Answer

(a) \( f'(x) = 1 - \frac{1}{x^2} \). (b) Graphs confirm that \( f' \) accurately describes \( f \)'s slope.

Step 1: Find the Derivative of Each Term

The function given is \( f(x) = x + \frac{1}{x} \). To find \( f'(x) \), we start by differentiating each term separately. The derivative of \( x \) is 1. For the term \( \frac{1}{x} \), we can rewrite it as \( x^{-1} \) and then differentiate. The derivative of \( x^{-1} \) is \( -x^{-2} \).

Step 2: Combine the Derivatives to Form \( f'(x) \)

Now combine the derivatives of each term: the derivative of \( x \) is 1, and the derivative of \( \frac{1}{x} \) is \( -x^{-2} \). Therefore, the derivative \( f'(x) = 1 + \left(-\frac{1}{x^2}\right) = 1 - \frac{1}{x^2} \).

Step 3: Verify by Graphing

To ensure that the calculated derivative \( f'(x) = 1 - \frac{1}{x^2} \) is reasonable, graph both \( f(x) = x + \frac{1}{x} \) and \( f'(x) \). The graph of \( f'(x) \) should represent the slope of \( f(x) \) at any given point \( x \). Notice that where \( f(x) \) increases, \( f'(x) \) should be positive, and where \( f(x) \) decreases, \( f'(x) \) should be negative. Also, observe that \( f'(x) = 0 \) at points where \( f(x) \) has a horizontal tangent.

Key Concepts

Differentiation Rules

In calculus, differentiation rules are essential for finding the rate at which a function changes with respect to its variable. This process is known as taking the derivative. In our exercise, we are tasked with finding the derivative of the function \( f(x) = x + \frac{1}{x} \). To do this, we apply differentiation rules to each term individually.

For the term \( x \), the derivative is quite simple: \( \frac{d}{dx}x = 1 \). This follows from the power rule, where the derivative of \( x^n \) is \( nx^{n-1} \), and in this case, \( n = 1 \).

Next, we deal with \( \frac{1}{x} \), which can be rewritten as \( x^{-1} \). Applying the power rule here, we differentiate to get \( -x^{-2} \), which is \( -\frac{1}{x^2} \).

Finally, we combine these results: the derivative of the entire function \( f(x) \) becomes \( f'(x) = 1 - \frac{1}{x^2} \). Applying differentiation rules allows us to systematically and accurately find the rate of change for various functions.

Graphing Functions

Graphing functions is a powerful way to visualize the behavior of a function and its derivative. When we graph our original function \( f(x) = x + \frac{1}{x} \) alongside its derivative \( f'(x) = 1 - \frac{1}{x^2} \), we can gain insight into how the function behaves at different points.

The graph of \( f(x) \) shows how the function increases or decreases over the domain. Where \( f(x) \) is increasing, its slope is positive, and \( f'(x) \) will also be positive. Conversely, where \( f(x) \) is decreasing, \( f'(x) \) is negative, indicating a negative slope.

A key takeaway is that whenever \( f'(x) = 0 \), \( f(x) \) has a horizontal tangent, indicating a local maximum or minimum. This happens because the slope of the tangent line is zero at these points, showing a transition in direction. Graphing functions and their derivatives is an excellent way to confirm analytical results and deepen understanding.

Calculus Problem Solving

Calculus problem solving involves applying different strategies and understanding key concepts of calculus, such as derivatives, to solve problems effectively. In our exercise, we used the concept of derivatives to analyze the function \( f(x) = x + \frac{1}{x} \).

By finding the derivative, \( f'(x) = 1 - \frac{1}{x^2} \), we determined how the function’s output changes with respect to its input. This information is crucial for solving various calculus problems, such as finding local maxima, minima, and points of inflection.

When engaging in calculus problem solving:

• Always identify the function and what you need to find, whether it’s a rate of change, an area under the curve, etc.
• Use differentiation rules to find derivatives when needed.
• Compare results by graphing to ensure that your analytical solutions are reasonable.

Problem solving in calculus is about connecting abstract mathematical concepts with practical analytical methods to draw meaningful conclusions from functions.