Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ f(x)=\alpha x+\beta $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \alpha x + \beta \) is \( f^{\prime}(x) = \alpha \).

Step 1: Setup the Limit Definition of Derivative

The derivative of a function \( f(x) \) at a point \( x \) is given by the limit definition: \[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\]Substitute \( f(x) = \alpha x + \beta \) into the formula.

Step 2: Compute \( f(x+h) \)

Substitute \( x + h \) into the function:\[f(x+h) = \alpha (x+h) + \beta = \alpha x + \alpha h + \beta.\]

Step 3: Substitute into the Limit Definition

Replace \( f(x+h) \) and \( f(x) \) into the limit expression:\[f^{\prime}(x) = \lim_{h \to 0} \frac{(\alpha x + \alpha h + \beta) - (\alpha x + \beta)}{h}.\]

Step 4: Simplify the Expression

Simplify the numerator:\[(\alpha x + \alpha h + \beta) - (\alpha x + \beta) = \alpha h.\]The expression becomes:\[f^{\prime}(x) = \lim_{h \to 0} \frac{\alpha h}{h}.\]

Step 5: Cancel \( h \) and Evaluate the Limit

Cancel \( h \) from the numerator and the denominator:\[f^{\prime}(x) = \lim_{h \to 0} \alpha = \alpha.\]Since \( h \) no longer appears in the expression, the limit directly evaluates to \( \alpha \).

Key Concepts

Limit Definition of Derivative

The limit definition of the derivative is the fundamental concept when it comes to understanding calculus. This definition provides a way to calculate the derivative of a function at a given point. The derivative represents the rate at which a function changes as its input changes, giving us insightful information about the function's behavior.
The mathematical expression for the derivative of a function \( f(x) \) is:

• \( f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

This expression involves taking the limit as \( h \) approaches zero. It uses the concept of a secant line—a line connecting two points on a curve—and examines what happens as the two points get infinitely close. In essence, the derivative at a point \( x \) is the slope of the tangent line to the curve at that point.
For the function \( f(x) = \alpha x + \beta \), this means examining how the equation changes as we apply an infinitesimally small increment \( h \) to \( x \). This setup helps, especially when working with linear functions.

Linear Function

A linear function is one of the simplest types of functions in algebra and calculus. It is represented in the general form \( f(x) = \alpha x + \beta \), where \( \alpha \) and \( \beta \) are constants.
Here:

• \( \alpha \) is the slope of the line.
• \( \beta \) is the y-intercept, where the line crosses the y-axis.

These functions are called "linear" because their graph is a straight line. The slope, \( \alpha \), tells you how steep the line is and determines the rate of change of the function.
For instance, if you have a linear function representing a scenario like speed as a function of time, the slope will tell you the speed at which that change is happening consistently over time, making calculations straightforward and predictable.

Differentiation Rules

Differentiation is a process in calculus that deals with finding the derivative of functions. Derivative rules provide shortcuts and formulas to simplify this process. There are standard rules that apply to linear functions, polynomials, products, quotients, and compositions of functions.
For a linear function like \( f(x) = \alpha x + \beta \), the differentiation rule is straightforward:

• The derivative of \( \alpha x \) is \( \alpha \), because the constant \( \alpha \) represents the constant rate of change.
• The derivative of a constant \( \beta \) is zero, since a constant term does not change as \( x \) changes.

Thus, the derivative of a linear function is simply the slope of the line, \( f^{\prime}(x) = \alpha \).
This shows why linear functions are so simple to differentiate. They reinforce the concept that the derivative of a line is essentially its slope, making linear equations a perfect starting point for learning differentiation.