Problem 29 - Calculus Single Variable

Question

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ (x-1) /(x+1) $$

Step-by-Step Solution

Answer

The derivative of \( \frac{x-1}{x+1} \) is \( \frac{2}{(x+1)^2} \).

Step 1: Identify the functions in the quotient

In the given expression \( \frac{(x-1)}{(x+1)} \), we have two functions: the numerator \( u(x) = x - 1 \) and the denominator \( v(x) = x + 1 \). To apply the Quotient Rule, we need to differentiate these functions separately.

Step 2: Differentiate the numerator and denominator

Differentiate the numerator \( u(x) = x - 1 \). The derivative is \( u'(x) = 1 \).Differentiate the denominator \( v(x) = x + 1 \). The derivative is \( v'(x) = 1 \).

Step 3: Apply the Quotient Rule

The Quotient Rule states that the derivative of \( \frac{u(x)}{v(x)} \) is given by: \[ (u/v)' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]Substitute \( u(x) = x - 1 \), \( u'(x) = 1 \), \( v(x) = x + 1 \), and \( v'(x) = 1 \):\[ = \frac{(1)(x+1) - (x-1)(1)}{(x+1)^2} \]

Step 4: Simplify the expression

Expand and simplify the numerator:\[ = \frac{x + 1 - x + 1}{(x+1)^2} \]Combine like terms:\[ = \frac{2}{(x+1)^2} \]

Step 5: State the derivative

The derivative of the given expression \( \frac{x-1}{x+1} \) is \( \frac{2}{(x+1)^2} \).

Key Concepts

Derivative

The concept of a derivative is foundational in calculus and refers to the rate at which a function is changing at any given point. Think of it as the slope of the function at a particular point on its curve. When you take the derivative of a function, you're essentially finding how the output of the function changes as the input changes minutely. Derivatives help in understanding the behavior and shape of the graph of a function, and they have a variety of applications, from physics to economics. In this problem, we are interested in finding the derivative of the quotient of two functions, which is where the quotient rule comes into play.

Numerator and Denominator Functions

To work with the quotient of two functions in calculus, you first need to identify and understand both the numerator and the denominator of the expression. The numerator is the upper part of a fraction, and the denominator is the lower part.

• For the given function \( \frac{x-1}{x+1} \), the numerator function is \( u(x) = x - 1 \).
• The denominator function is \( v(x) = x + 1 \).

Identifying these functions is vital because applying the quotient rule requires knowing how to differentiate each one separately. This structured approach allows us to methodically find the derivative of the entire expression.

Differentiation

Differentiation is the process of finding the derivative of a function. For straightforward functions like polynomials, differentiation can be simple. You essentially need to apply basic differentiation rules, such as the power rule or the constant rule.

• For the numerator \( u(x) = x - 1 \): The derivative is \( u'(x) = 1 \) because the derivative of \( x \) is \( 1 \) and the derivative of a constant is \( 0 \).
• Similarly, for the denominator \( v(x) = x + 1 \): The derivative is \( v'(x) = 1 \).

These derivatives are then used in the quotient rule to differentiate the entire quotient.

Simplifying Rational Expressions

After applying the quotient rule to find the derivative of a fraction, the result can often be a complex expression. Simplifying rational expressions is crucial because it makes the results more interpretable and usable for further calculations.

• With the expression derived from the quotient rule: \( \frac{1(x+1) - (x-1)(1)}{(x+1)^2} \), the goal is to simplify the numerator.
• Expanding the terms gives \( x + 1 - x + 1 \).
• This simplifies to \( 2 \), because \( x - x \) cancels out.

Thus, the simplified derivative of the original expression is \( \frac{2}{(x+1)^2} \). Simplifying makes it clearer and easily applicable for practical uses.