Problem 65 - Brief Applied Calculus

Question

Explain, in two different ways, without using the rules of differentiation, why the derivative of the constant function \(f(x)=2\) must be \(f^{\prime}(x)=0\). [Hint: Think of the slope of the graph of a constant function, and also of the instantaneous rate of change of a function that stays constant.]

Step-by-Step Solution

Answer

The derivative of a constant function is zero because its slope is flat and its rate of change is null.

Step 1: Understanding the Slope of a Constant Graph

Consider the function \(f(x) = 2\), which is a horizontal line on the graph. A horizontal line has no inclination, meaning its slope is zero because the rise over run ratio is zero since there is no vertical change. Hence, the derivative, which is the slope of the function at any point, is zero: \(f'(x) = 0\).

Step 2: Rate of Change for a Constant Function

The derivative is essentially a measure of how the function value changes as \(x\) changes. For \(f(x) = 2\), this value does not change regardless of the value of \(x\). The constant '2' means the rate of change (difference in 'y' values for any change in 'x') is always zero, hence \(f'(x) = 0\).

Key Concepts

Slope of a Horizontal Line

The slope of a line is a measure of its steepness or incline. For a horizontal line, like the one described by the constant function \( f(x) = 2 \), the line runs parallel to the x-axis. This means that no matter how far you move along the x-axis, the y-value, which stays constant at 2, does not change.

The mathematical way to understand slope is through the formula "rise over run," which is the change in y divided by the change in x. With a horizontal line, there is no rise; the change in y is zero. Consequently, the slope is zero. Hence, the derivative of a constant function \( f(x) \) is\( f'(x) = 0 \), because it represents the slope at any given point on the line.

• Horizontal line means no vertical change.
• Slope = \( \frac{0}{\text{change in } x} = 0 \).
• Derivative of constant function is zero.

Instantaneous Rate of Change

A derivative can be thought of as the instantaneous rate of change of a function. This concept can be compared to how fast a car is moving at a specific moment. For the constant function \( f(x) = 2 \), this measurement looks at how the output (y-value) changes as x changes.

In the case of \( f(x) = 2 \), the y-value does not change at all; it is always the same, irrespective of the x-value. This signifies that at any point, the rate at which the function changes is zero. In essence, the value is static and not changing over time, and therefore, the derivative \( f'(x) \) is again zero.

• Function output remains constant as input changes.
• No change in function implies a rate of change of zero.
• Derivative measures this "no change," hence is zero.

Rules of Differentiation

The rules of differentiation, such as the power rule, product rule, and chain rule, are systematic methods for finding derivatives of various functions. However, the simplicity of differentiating a constant reveals that these rules align perfectly with logical reasoning about constant functions.

When using the rules of differentiation, a constant function \( f(x) = c \), where \( c \) is any real number, always results in a derivative of zero. This fits perfectly because the rules dictate that being constant implies no change, and hence the derivative, as a measure of change, is zero. Even though we didn't use these rules in the exercise, knowing them can provide a structured approach in more complex situations.

• Constant derivative rule: \( \frac{d}{dx}[c] = 0 \).
• Illustrates why derivatives of unmatched functions are zero.
• Provides a foundation for approaching more complex derivatives.