Problem 26 - 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). $$ f(x)=\frac{x+3}{x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\frac{3}{x^2} \).

Step 1: Identify f(t) and f(x)

Given the function \( f(x) = \frac{x+3}{x} \), we can replace \( x \) with \( t \) to write \( f(t) = \frac{t+3}{t} \). Hence, \( f(x) = \frac{x+3}{x} \).

Step 2: Set Up the Difference Quotient

Substitute \( f(t) \) and \( f(x) \) into the limit definition of the derivative: \( f'(x) = \lim _{t \rightarrow x} \frac{f(t) - f(x)}{t-x}\). This becomes \( \lim_{t \rightarrow x} \frac{\frac{t+3}{t} - \frac{x+3}{x}}{t-x} \).

Step 3: Subtract the Fractions

To subtract fractions \( \frac{t+3}{t} - \frac{x+3}{x} \), find a common denominator, which is \( xt \). This gives us \( \frac{(t+3)x - (x+3)t}{xt} = \frac{(xt + 3x - xt - 3t)}{xt} = \frac{3x - 3t}{xt} \).

Step 4: Simplify the Numerator

Factor out a 3 from the numerator to get \( \frac{3(x - t)}{xt} \). This allows \( x-t \) to cancel with the denominator of the difference quotient, \( t-x \), leaving \( -\frac{3}{xt} \) because \( x-t = -(t-x) \).

Step 5: Apply the Limit

Now apply the limit \( \lim_{t \rightarrow x} -\frac{3}{xt} \). As \( t \rightarrow x \), \( t \) approaches \( x \), so \( xt \rightarrow x^2 \). Therefore, the limit becomes \( -\frac{3}{x^2} \).

Step 6: Final Derivative

From the steps above, the derivative \( f'(x) \) is \( -\frac{3}{x^2} \).

Key Concepts

Limit Definition of Derivative

The limit definition of a derivative is foundational in calculus. It is a mathematical way to describe the rate at which a function value changes as its input changes. In its basic form, the derivative of a function \( f(x) \) at a point \( x \) is given by:
\[f'(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}\]This expression is often conceptualized as the "instantaneous rate of change" of the function at a particular input value. It captures the essence of how a function behaves exactly at a point, rather than over an interval.

• The term \( \lim_{t \to x} \) indicates that as \( t \) gets infinitely close to \( x \), we are observing how the quotient behaves.
• The numerator \( f(t) - f(x) \) represents the change in function values.
• The denominator \( t-x \) represents the change in input values.

Understanding this limit helps in grasping how derivatives represent slopes of tangents to curves, a critical concept in topics ranging from physics to economics.

Difference Quotient

The difference quotient is an expression used in the process of finding a derivative using the limit definition. It can initially be expressed as:
\[\frac{f(t) - f(x)}{t - x}\]This expression looks at the average rate of change of the function between two points, \( t \) and \( x \). The goal is to then convert this average rate of change into an instantaneous rate of change through the application of a limit.

• It is the backbone of the derivative, expressing the gradient or slope of a secant line connecting two points on the function curve.
• As we refine this expression using limits, we transition from the secant slope to the tangent slope, the key goal of derivative calculation.

In the context of rational functions, which are ratios of polynomials, working with the difference quotient often involves complex fraction manipulation. This can include finding a common denominator, simplifying the resulting expression, and factoring as seen in our example.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They are expressed in the form:
\[f(x) = \frac{P(x)}{Q(x)}\]where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are ubiquitous in mathematics due to their versatility and applicability to various fields.

• The operations involving rational functions, like determining the derivative, can be complex due to the need for managing polynomial division or subtraction and finding common denominators.
• Analyzing limits and behavior as \( x \) approaches certain values often requires techniques such as factorization and simplification.

In our derivative example, the polynomial expression in rational functions led to steps involving common denominators—essential for simplifying complex fractions and revealing the derivative. Managing these manipulations carefully reflects the systematic approach needed for working with rational functions in calculus.