Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=\frac{x}{x-5} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{5}{(x-5)^2}\).

Step 1: Identify the function and its components

We are given the function \( f(x) = \frac{x}{x-5} \). We need to find its derivative \( f^{\prime}(x) \) using the limit definition of derivative.

Step 2: Express the difference quotient

According to the limit definition of the derivative, we write the difference quotient: \[ \frac{f(t) - f(x)}{t-x} = \frac{\frac{t}{t-5} - \frac{x}{x-5}}{t-x}. \]

Step 3: Find a common denominator

To simplify the difference quotient, find a common denominator for \( \frac{t}{t-5} \) and \( \frac{x}{x-5} \). The common denominator is \((t-5)(x-5)\).

Step 4: Simplify the numerators

Rewriting the fractions: \( \frac{t}{t-5} = \frac{t(x-5)}{(t-5)(x-5)} \) and \( \frac{x}{x-5} = \frac{x(t-5)}{(t-5)(x-5)} \). Subtract these to get: \[ \frac{t(x-5) - x(t-5)}{(t-5)(x-5)(t-x)}. \]

Step 5: Expand and simplify the numerator

Expand the expression in the numerator: \( t(x-5) - x(t-5) = tx - 5t - xt + 5x \). Simplify this to \( 5x - 5t \) or \( 5(x-t) \).

Step 6: Factor and simplify the expression

Substitute the simplified numerator back into the fraction to get: \[ \frac{5(x-t)}{(t-5)(x-5)(t-x)}. \] Notice that \(x-t\) and \(t-x\) are negatives of each other, so they cancel, leaving \( -5 \).

Step 7: Apply the limit

Therefore, the limit becomes: \[ f^{\prime}(x) = \lim_{t \to x} -\frac{5}{(t-5)(x-5)}. \] After canceling the terms, and as \(t\to x\), the result simplifies to \[ f^{\prime}(x) = -\frac{5}{(x-5)^2}. \]

Key Concepts

Limit Definition of Derivative

To understand how to find the derivative using the limit definition, it's important to start with the fundamental idea of what a derivative is. The derivative, denoted as \( f^{\prime}(x) \), provides the rate at which a function \( f(x) \) changes at a certain point. One of the common ways to compute a derivative is through the limit definition:

• \( f^{\prime}(x) = \lim_{t \rightarrow x} \frac{f(t) - f(x)}{t-x} \)

This expression essentially considers an infinitesimally small change around a point \( x \) and how the function value at \( x \) responds to this change.
For our function \( f(x) = \frac{x}{x-5} \), we're looking to calculate \( f^{\prime}(x) \) using this definition. The core idea is substituting into this limit form, then simplifying the expression without getting sidetracked by algebraic challenges.

Rational Functions

Rational functions are expressions that involve the division of two polynomials. A function like \( f(x) = \frac{x}{x-5} \) is a simple rational function because it consists of a linear function in the numerator and a linear function in the denominator. When dealing with derivatives of rational functions, especially using the limit definition, it's vital to handle the algebraic complexity carefully.
Rational functions often have discontinuities, which occur where the denominator equals zero. In this case, it would be at \( x=5 \), where the function is undefined. Understanding this helps avoid mistakes when simplifying expressions. Additionally, simplifying rational expressions requires diligent factoring and finding common terms, which are key parts of calculating derivatives.

Common Denominator

When using the limit definition to find the derivative of a function like \( \frac{x}{x-5} \), we're often left needing to subtract fractions. To subtract such fractions, it's essential to find a common denominator.

• In our example, we have \( \frac{t}{t-5} \) and \( \frac{x}{x-5} \).
• The common denominator for these two terms is \((t-5)(x-5)\).

Finding the common denominator allows us to rewrite each fraction with this denominator, facilitating subtraction and further simplification. Determining the least common denominator is not only essential for simplifying, but it's also crucial for correctly applying the limit definition of a derivative.

Simplifying Expressions

In working through the derivative using limits, simplifying expressions is often one of the most challenging tasks. We initially find the difference quotient that needs simplification:

• \( \frac{t(x-5) - x(t-5)}{(t-5)(x-5)(t-x)} \)

By expanding and combining like terms in the numerator, we simplify it to \( 5(x-t) \). At this stage, it's vital to recognize that \( (x-t) \) and \( (t-x) \) are additive inverses. This insight lets us cancel terms that yield \(-5\), leading us to a much simpler expression.
Finally, applying the limit as \( t \) approaches \( x \), any terms directly involving \( t \) in the numerator vanish, leaving an elegant expression for the derivative. This process of expanding, combining terms, and canceling is crucial for mastering derivatives by the limit method.