Problem 23 - Precalculus Mathematics 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: Identify the function

The function given is \( f(x) = \frac{x}{x+1} \). We'll need to differentiate this function with respect to \( x \) to find the derivative \( f'(x) \).

Step 2: Apply the Quotient Rule

The quotient rule is used for differentiating functions that are ratios of two differentiable functions. The quotient rule states: \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). Here, \( u = x \) and \( v = x+1 \).

Step 3: Find the derivatives of the numerator and the denominator

Calculate the derivative of the numerator: \( u' = x' = 1 \). Calculate the derivative of the denominator: \( v' = (x+1)' = 1 \).

Step 4: Substitute into the Quotient Rule

Substitute the derivatives and original functions into the quotient rule formula:\[ f'(x) = \frac{(1)(x+1) - (x)(1)}{(x+1)^2} \]

Step 5: Simplify the expression

Simplify the expression obtained in Step 4:\[ f'(x) = \frac{x + 1 - x}{(x+1)^2} = \frac{1}{(x+1)^2} \]

Step 6: Substitution of \(a\) into the derivative

Replace \( x \) with \( a \) in the derivative to find \( f'(a) \):\[ f'(a) = \frac{1}{(a+1)^2} \]

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus used to differentiate rational functions, which are functions formed as the ratio of two differentiable functions. When you encounter a function like

• \( f(x) = \frac{u}{v} \),
• where both \( u \) and \( v \) are differentiable,

you apply the quotient rule to find the derivative.

The quotient rule simplifies to:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]It essentially tells us that to differentiate a quotient, you multiply the derivative of the numerator \( u' \) by the denominator \( v \), subtract the product of the numerator \( u \) and the derivative of the denominator \( v' \), and then divide the result by the square of the denominator \( v^2 \).

This rule is especially handy for rational functions, which is our next topic.

Derivative

A derivative represents the rate at which a function is changing at any given point. It is a fundamental concept:

• The derivative of a function \( f(x) \) is denoted as \( f'(x) \) or \( \frac{df}{dx} \).
• It gives the slope of the tangent line to the function at any point.

Understanding how to find derivatives allows you to understand how functions behave, and it's used across various fields.

In our exercise, the function \( f(x) = \frac{x}{x+1} \) required us to use the quotient rule to find its derivative. First, we identified:

• The numerator \( u = x \) and its derivative \( u' = 1 \).
• The denominator \( v = x+1 \) and its derivative \( v' = 1 \).

Finally, by applying all these into the quotient rule, the derivative of the function was successfully calculated.

Rational Functions

Rational functions are crucial in mathematics. They are essentially functions that can be expressed as the ratio of two polynomial functions. In more formal terms:

• A rational function \( f(x) \) is given by \( \frac{p(x)}{q(x)} \),
• where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \).

These functions have various applications in fields such as physics, engineering, and economics due to their nature of representation.

For our specific problem, \( f(x) = \frac{x}{x+1} \) is a classic example of a rational function:

• The numerator being \( x \), a simple linear polynomial,
• The denominator being \( x+1 \), another linear polynomial.

When differentiating rational functions like this one, the quotient rule is your best friend, as it makes tackling these quotients straightforward and systematic.