Problem 34 - Calculus and its applications

Question

Find \(f^{\prime}(x)\). $$ f(x)=\frac{2 x}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The given function is \( f(x) = \frac{2x}{3} \). This is a linear function because it can be rewritten as \( f(x) = \frac{2}{3}x \) which is in the form \( ax + b \), where \( a = \frac{2}{3} \) and \( b = 0 \).

Step 2: Recall the Derivative of a Linear Function

The derivative of a linear function \( f(x) = ax + b \) is simply \( a \). This is because the derivative measures the rate of change, and for a linear function, this rate of change is constant and equals the coefficient of \( x \).

Step 3: Apply the Derivative Rule

Using the rule from the previous step, the derivative of \( f(x) = \frac{2}{3}x \) is simply the coefficient of \( x \), which is \( \frac{2}{3} \). Thus, \( f'(x) = \frac{2}{3} \).

Key Concepts

Understanding Linear Functions

A linear function is one of the simplest types of functions in mathematics. It is commonly expressed in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. Linear functions graph as straight lines and have a constant slope.

To visualize it:

• \( a \) determines the slope or steepness of the line.
• \( b \) represents the y-intercept, which is where the line crosses the y-axis.

Linear functions are straightforward yet foundational as they form the basis for more complicated functions in calculus and algebra. A key characteristic is their constant rate of change, which stays the same regardless of which two points you pick on the line.

Exploring the Rate of Change

The rate of change of a function tells us how much the output value changes for a given change in the input value. For linear functions, this rate of change is constant and directly equals the slope \( a \) in the expression \( f(x) = ax + b \).

Here's what you need to know:

• In the context of a linear function, the rate of change is uniform, meaning the function increases or decreases steadily.
• This is why the derivative of a linear function, which measures this rate, remains constant.

Understanding the rate of change is crucial because it simplifies the process of finding derivatives in linear functions. It gives insight into the behavior of functions and prepares you for more complex topics like slopes of tangent lines in calculus.

Importance of the Coefficient of x

The coefficient of \( x \) in a linear function \( f(x) = ax + b \) plays a vital role in determining the function's behavior. This coefficient influences both the slope of the line on a graph and the derivative of the function.

When talking about derivatives:

• The derivative of a linear function \( f(x) = ax + b \) is \( f'(x) = a \). This derivative represents the function's slope or rate of change.
• The derivative is constant because the coefficient \( a \) remains the same across any interval of \( x \).

By understanding the coefficient of \( x \), you can easily determine key attributes of linear functions, helping to quickly solve problems and interpret graphs.