Problem 21 - Precalculus: Mathematical for Calculus

Question

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=\frac{x}{x+1}$$

Step-by-Step Solution

Answer

\( f'(a) = \frac{1}{(a+1)^2} \)

Step 1: Understand the Derivative Definition

To find the derivative of a function at a given point, we need to compute \( f'(a) = \frac{d}{dx} \left. f(x) \right|_{x = a} \). This involves finding the derivative of \( f(x) \) first.

Step 2: Use the Quotient Rule

The function \( f(x) = \frac{x}{x+1} \) is a quotient of two functions. The quotient rule for derivatives is \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \), where \( u = x \) and \( v = x + 1 \).

Step 3: Differentiate the Numerator and Denominator

Find \( u' \) and \( v' \) where \( u = x \) and \( v = x+1 \). Thus, \( u' = 1 \) and \( v' = 1 \).

Step 4: Apply the Quotient Rule

Using the quotient rule, \( f'(x) = \frac{(1)(x + 1) - (x)(1)}{(x+1)^2} = \frac{x + 1 - x}{(x+1)^2} = \frac{1}{(x+1)^2} \).

Step 5: Evaluate the Derivative at \( x = a \)

Now substitute \( x = a \) into \( f'(x) \) to find \( f'(a) \). This results in \( f'(a) = \frac{1}{(a+1)^2} \).

Key Concepts

Quotient Rule

When dealing with derivatives, you often encounter functions expressed as one function divided by another. The quotient rule is a handy tool for differentiating such functions. It allows you to find the derivative of a fraction where both the numerator and the denominator are themselves functions of a variable. This rule states that if you have a function \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is given by:

• \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

This formula may seem complex at first, but it's essentially a method to ensure that the derivative accounts for changes in both the numerator and denominator.

The important steps include:

• Finding the derivative of the numerator \( u' \).
• Finding the derivative of the denominator \( v' \).
• Plugging these derivatives into the quotient rule formula.

Once all components are in place, carrying out basic algebra simplifies the expression to find \( f'(x) \). Applying the quotient rule correctly ensures you accurately obtain the derivative of fraction-based functions.

Function Differentiation

Function differentiation is the process of finding the rate at which a function changes at any given point. Mathematically, it's all about discovering how a small change in input impacts the output. Consider the function \( f(x) = \frac{x}{x+1} \). Here, both the numerator \( x \) and the denominator \( x + 1 \) are functions of \( x \).

• The numerator \( u = x \) has a derivative \( u' = 1 \).
• The denominator \( v = x + 1 \) also has a derivative \( v' = 1 \).

Differentiating these individual parts is the first crucial step before applying any rule like the quotient rule.

By understanding the differentiation of each function part independently, you can piece them together using rules like the quotient rule to evaluate the overall derivative. This makes complex differentiations manageable.

Derivative Evaluation

Once you've derived the formula using differentiation techniques and rules such as the quotient rule, the final step is to evaluate it at a specific point. For our function \( f(x) = \frac{x}{x+1} \), after applying the quotient rule, we derived \( f'(x) = \frac{1}{(x+1)^2} \).

• The derivative function \( f'(x) \) is simplified to express the instantaneous rate of change of the function at any point \( x \).
• To find the derivative at a particular value \( a \), simply substitute \( x = a \) into the derived formula.

This results in \( f'(a) = \frac{1}{(a+1)^2} \), giving you the exact rate of change at \( x = a \).

Evaluating the derivative accurately allows you to understand how the function behaves at specific points, providing valuable insights into the function's growth or decay dynamics.