Problem 40 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$ f(x)=4 x-7 $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = 4x - 7 \) is \( f^{ ext{"}}(x) = 4 \).

Step 1: Identify the Function

We begin by identifying the given function. The function is given as \( f(x) = 4x - 7 \). This is a linear function.

Step 2: Recall Differentiation Rules

Remember that the derivative of \( ax + b \), where \( a \) and \( b \) are constants, is simply \( a \). The differentiation rule for a constant \( b \) is that its derivative is 0 because constants do not change.

Step 3: Differentiate the Function

Apply the differentiation rules to find the derivative of the function. The derivative of \( 4x \) with respect to \( x \) is \( 4 \). The derivative of \( -7 \), a constant, is \( 0 \). Thus, the derivative of the entire function \( f(x) = 4x - 7 \) is \( 4 \).

Step 4: Write the Result

Combine the derivatives to write the final result. Thus, \( f^{ ext{"}}(x) = 4 \) is the derivative of the original function.

Key Concepts

Linear Function

A linear function is one of the simplest forms of functions you can encounter in mathematics. It is typically expressed as \( f(x) = ax + b \), where \( a \) and \( b \) are constants. Linear functions are called "linear" because their graph is a straight line, the most basic form of a line graph.
These functions are widely used because they have a constant slope, which makes them predictable and easy to understand. The parameter \( a \) represents the slope, which tells us how steep the line is, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.
For example, in the function \( f(x) = 4x - 7 \), the slope \( a \) is 4, suggesting the line rises 4 units for every 1 unit it moves to the right. The y-intercept \( b \) is -7, indicating that when \( x = 0 \), \( f(x) \) will be -7. Understanding linear functions is crucial as they serve as a foundation upon which many more complex mathematical concepts are built.

Derivative Rules

Differentiation is a fundamental concept in calculus, primarily used to determine how a function changes at any given point. The main goal is to find a function's derivative, which essentially gives the slope of the function’s curve at any point along it.
To master differentiation, it's vital to familiarize yourself with some basic derivative rules:

• The **constant rule** dictates that the derivative of a constant function is zero because constants have no rate of change.
• The **power rule** states that the derivative of \( x^n \) is \( nx^{n-1} \), which helps simplify finding derivatives of polynomial functions.
• The **constant multiplier rule** allows us to pull constants out of the differentiation operation, so the derivative of \( c\cdot f(x) \) is \( c \cdot f^{\prime}(x) \).
• The **sum rule** lets us differentiate each term separately when finding the derivative of a sum, so \( (f + g)^{\prime} = f^{\prime} + g^{\prime} \).

In our example, \( f(x) = 4x - 7 \), we utilize these rules to find \( f^{\prime}(x) \). The derivative of \( 4x \) using the constant multiplier rule is 4, and the derivative of the constant -7 is 0, simplifying \( f^{\prime}(x) \) to 4.

Constant Function Differentiation

When dealing with the differentiation of constant functions, it’s straightforward yet essential. A constant function is written simply as \( f(x) = c \), where \( c \) is a constant number.
The derivative of a constant function is always zero because a constant doesn't change. For example, if a function \( f(x) \) always equals 7, no matter what \( x \) is, then it has no variation, and hence its rate of change is zero.
In the context of our example \( f(x) = 4x - 7 \), the term \(-7\) highlights this constant aspect. By applying the derivative rules, we see that the derivative of \(-7\) is indeed 0.
Understanding that constants contribute no slope reinforces why linear functions like our example are straightforward to differentiate. Each term is considered separately, and the non-changing value is acknowledged as having a derivative of zero, leaving focus on the terms with variable components. This makes handling linear functions both efficient and intuitive.