Question

Multiple Choice Which of the following is \(\frac{d}{d x}\left(\frac{x+1}{x-1}\right) ?\) (A) \(\frac{2}{(x-1)^{2}} \quad\) (B) 0\(\quad\) (C) \(-\frac{x^{2}+1}{x^{2}}\) (D) \(2 x-\frac{1}{x^{2}}-1 \quad\) (E) \(-\frac{2}{(x-1)^{2}}\)

Short answer

Option A is the correct answer, i.e \(\frac{2}{(x-1)^{2}} \)

Step-by-step solution

Find the derivative of top function (nominator)

The derivative of the function \( f(x) = x + 1 \) is \( f'(x) = 1 \) because the derivative of constant 1 is 0 and the derivative of x is 1.

Find the derivative of bottom function (denominator)

The derivative of the function \( g(x) = x - 1 \) is \( g'(x) = 1 \) because the derivative of constant -1 is 0 and the derivative of x is 1.

Apply the Quotient Rule

Now, form the derivative using Quotient Rule. \( f'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \), if you substitute \( f'(x) = 1, g'(x) = 1, f(x) = x+1 , g(x) = x-1 \) in the above function, the derivative will be \( f'(x) = \frac{1*(x-1) - (x+1)*1}{(x-1)^2} = \frac{2}{(x-1)^2} \)

Key concepts

Quotient Rule

In calculus, the Quotient Rule is a fundamental technique used to determine the derivative of a ratio of two functions. It's especially handy when you're trying to find the derivative of a rational function, which is simply one function divided by another. According to the rule, if you have a function in the form of \(f(x) = \frac{g(x)}{h(x)}\), where both \(g(x)\) and \(h(x)\) are differentiable and \(h(x) \eq 0\), their derivative \(f'(x)\) is given as: \[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2} \]

Essentially, you take the derivative of the numerator function \(g'(x)\) and multiply it by the denominator function \(h(x)\), and from that product, you subtract the product of the numerator function \(g(x)\) and the derivative of the denominator function \(h'(x)\). Then, you divide the entire expression by the square of the denominator function \(h(x)^2\). This is a critical tool for solving many calculus problems where functions are divided by one another.

Derivative of Rational Function

When working with derivatives of rational functions, it's important to recognize that a rational function is simply a fraction where both the numerator and denominator are polynomials. To differentiate a rational function, you can use the Quotient Rule as explained earlier, or in certain situations, you might be able to simplify the function before taking the derivative.

Keep in mind that before applying the Quotient Rule, you should check for possibilities to simplify the fraction by factoring and canceling common terms. However, when simplification is not possible or practical, you'll need to use the Quotient Rule directly on the original function. It's important to perform each step carefully, as mistakes in differentiation can lead to incorrect results.

Calculus Problems

Solving calculus problems often involves understanding and applying different rules of differentiation, such as the Power Rule, Chain Rule, and the Quotient Rule. Difficulty arises when deciding which rule to apply, and how to apply it correctly. For many students, practicing these types of problems is key to mastery.

In the case of the example problem, we have a rational function, for which the Quotient Rule is the appropriate differentiation technique. Once you become familiar with various types of functions and their typical derivatives, you'll more quickly and accurately be able to identify the methods needed to solve calculus problems efficiently.

Derivative Calculations

Performing derivative calculations is a major part of calculus. Such calculations require understanding of various differentiation rules and the ability to apply them correctly. Every function type, whether it's polynomial, rational, exponential, or trigonometric, follows specific rules when being differentiated.

In the context of rational functions as seen in the given exercise, using the Quotient Rule accurately is crucial for successful derivative calculations. One must carefully apply each step: finding the derivatives of both the numerator and the denominator functions separately, and then combining them as per the rule's formula. With practice, derivative calculations become more intuitive and less error-prone. Guidance, such as that provided by step-by-step solutions, helps students check their work and understand the process behind each derivative calculation.