Problem 43 - A Graphical Approach to Precalculus with Limits

Question

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=3 x-1$$

Step-by-Step Solution

Answer

The derivative \(f'(a)\) is 3.

Step 1: Understanding the Function

We start with the linear function given by \(f(x) = 3x - 1\). This is a straightforward linear function of the form \(ax + b\), where \(a = 3\) and \(b = -1\).

Step 2: Recall the Derivative Rule for Linear Functions

For a linear function \(f(x) = ax + b\), the derivative \(f'(x)\) is simply the coefficient of \(x\), which is \(a\). This comes from the rule that the derivative of \(x\) is 1, so \(\frac{d}{dx}(ax + b) = a\).

Step 3: Differentiate the Function

Apply the derivative rule: since \(f(x) = 3x - 1\), the derivative \(f'(x)\) is \(3\).

Step 4: Evaluate the Derivative at a Specific Point

To find \(f'(a)\), substitute \(a\) into the expression for \(f'(x)\). It remains the constant \(3\) since the derivative of a linear function is constant.

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions you can encounter in mathematics. It has the general form of \(f(x) = ax + b\), where \(a\) and \(b\) are constants.
This type of function graphs as a straight line, with the slope of the line being \(a\). The constant term \(b\) represents the y-intercept, which is the value of the function when \(x = 0\).

• Example: The function \(f(x) = 3x - 1\) has a slope of \(3\) and a y-intercept at \(y = -1\).
• Slope: This number \(a\) shows the steepness of the line. A larger absolute value of \(a\) means a steeper slope.
• Intercept: The value \(b\) shifts the line up or down on the y-axis, determining where the line crosses the y-axis.

Derivative Rules

Derivative rules are guidelines that help us find the derivatives of various functions easily and quickly.
The rules simplify the process so that you don't need to go back to the most basic definition of a derivative every time.

• For Linear Functions: The derivative of a linear function \(f(x) = ax + b\) is \(f'(x) = a\). This means that the derivative of any linear function is its slope \(a\).
• Constant Rule: If you have a constant function \(f(x) = c\), its derivative is \(0\) because it remains unchanged as \(x\) changes.
• Power Rule: For a function \(x^n\), the derivative is \(nx^{n-1}\). This rule doesn't directly apply to linear functions without adjustment, since a linear function is a first-degree polynomial (\(n=1\)).

For our function \(f(x) = 3x - 1\), the derivative rule for linear functions tells us that \(f'(x) = 3\). This is a simplified process compared to more complex functions.

Evaluation of Derivatives

Evaluating derivatives involves applying the derivative you've found to particular values of \(x\). For linear functions, this is straightforward.
Since the derivative of a linear function is constant, evaluating it at any point \(a\) will yield the same result.
When given the function \(f(x) = 3x - 1\), its derivative is \(f'(x) = 3\). This means:

• Regardless of the value of \(a\) we choose, \(f'(a) = 3\).
• The derivative tells us that the slope of the tangent to the line at any point is always \(3\).

Thus, the evaluation of the derivative simply confirms the constant rate of change of the function. This concept is core to understanding how derivatives work in relation to linear functions.