Problem 48 - Contemporary Precalculus

Question

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=3 x-5$$

Step-by-Step Solution

Answer

Answer: The derivative of the function \(f(x) = 3x - 5\) is \(f'(x) = 3\).

Step 1: Identify the function parts

First, we need to identify the individual parts of the given function \(f(x) = 3x - 5\). This is a simple linear function with two terms: \(3x\) and \(-5\).

Step 2: Apply the power rule to each part

Now, we will apply the power rule to each individual part of the function. For the first term, \(3x\), we have: \(n = 1\) (the exponent of x) Applying the power rule, we get: \(3x^1 \to 1\cdot3x^{1-1} = 3x^0 = 3\) For the second term, \(-5\), we can rewrite it as \(-5x^0\), since any number raised to the power of 0 is 1. \(n = 0\) (the exponent of non-existent x) Applying the power rule, we get: \(-5x^0 \to 0\cdot(-5)x^{-1} = 0\)

Step 3: Combine the results of each part

Now, we will combine the results of the derivatives of each individual part to find the rule of the derivative of the function. Derivative of \(f(x)\): \(f'(x) = 3 - 0 = 3\) Thus, the rule of the derivative of the function \(f(x) = 3x - 5\) is \(f'(x) = 3\).

Key Concepts

Power Rule

The power rule is a fundamental technique in calculus used to find the derivative of functions that are in the form of a power of a variable, usually expressed as \(x^n\). This rule simplifies the differentiation process considerably, particularly when the function involves simple powers. The formula for the power rule is:

• Given a function \(f(x) = x^n\), the derivative \(f'(x) = n \cdot x^{n-1}\).

For instance, if you take a term like \(x^3\), applying the power rule would mean multiplying by the exponent (3 in this case) and reducing the exponent by 1, resulting in \(3 \cdot x^{2}\). For linear terms like \(3x\), treat it as \(3x^1\); therefore, the derivative is \(3 \cdot x^{0} = 3\). This rule collapses beautifully with constants or terms where \(n = 0\), resulting in zero because derivative of a constant is always zero as shown in \(-5x^0\).

Linear Function

A linear function is expressed as \(f(x) = mx + b\), where \(m\) and \(b\) are constants. In simpler terms, these functions graph as straight lines, with \(m\) representing the slope and \(b\) the y-intercept. For example, in the function \(f(x) = 3x - 5\), the slope \(m\) is 3, and the y-intercept is \(-5\). Linear functions are crucial in calculus and algebra as they provide a straightforward foundation for more complex functions.
One significant property of linear functions is that their derivative is a constant. This means the rate of change of linear functions is uniform, illustrated by a consistent slope. For \(f(x) = 3x - 5\), the derivative is \(f'(x) = 3\); the function's rate of change does not vary, reflecting the constant slope of the line.

Differentiation

Differentiation is the process of finding the derivative of a function. It allows us to understand how a function changes at any point.
The derivative of a function is essentially its "slope", describing how the function's value moves as its input changes. There's a whole field of calculus centered around this idea, as it is critical in understanding dynamic behaviors in models. Differentiation is vital across disciplines such as physics, economics, and engineering for analyzing varying rates.

• Finding derivatives involves applying rules like the power rule, product rule, or chain rule, depending on the function complexity.
• For simple polynomials and linear functions, as seen in the problem, efforts reduce to applying the power rule.

In essence, differentiation lets us peek into the immediate rate of change within any function, hence predicting behaviors and tendencies linked to these mathematical models.