Problem 54 - Brief Applied Calculus

Question

For each function: a. Find \(f^{\prime}(x)\) using the definition of the derivative. b. Explain, by considering the original function, why the derivative is a constant. $$ \begin{array}{l} f(x)=b\\\ (b \text { is a constant }) \end{array} $$

Step-by-Step Solution

Answer

The derivative \( f'(x) \) of the constant function \( f(x) = b \) is 0 because the function is constant and does not change with \( x \).

Step 1: Recall the Definition of the Derivative

The derivative of a function \( f(x) \) is defined as the limit: \[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \] This expression will help us calculate \( f'(x) \) for the given function.

Step 2: Substitute the Function into the Derivative Definition

For the constant function \( f(x) = b \), where \( b \) is a constant, substitute into the derivative definition:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} = \lim_{{h \to 0}} \frac{{b - b}}{h} \] Since \( f(x+h) = b \) and \( f(x) = b \), the difference \( b - b = 0 \).

Step 3: Simplify and Evaluate the Limit

The fraction becomes:\[ \frac{0}{h} = 0 \] Therefore, the limit evaluates to:\[ f'(x) = \lim_{{h \to 0}} 0 = 0 \]This means the derivative of the constant function \( f(x) = b \) is 0.

Step 4: Explain Why the Derivative is a Constant

For a constant function, the output value \( b \) does not change regardless of \( x \). Consequently, the rate of change of the function with respect to \( x \) is 0, which is why the derivative \( f'(x) \) is a constant 0.

Key Concepts

Derivative

In calculus, the derivative represents how a function changes as its input changes. It provides the rate at which the output of a function changes regarding an infinitesimally small change in input. When we talk about differentiating a function, we're essentially talking about finding this rate of change.

• The derivative of a function \( f(x) \) at a point \( x \) is denoted as \( f'(x) \) or \( \frac{df}{dx} \).
• We often find derivatives using the limit definition, which involves examining the function’s behavior as the change in its input approaches zero.

Understanding derivatives is essential because they help describe various phenomena like motion, growth, and change.
In this specific exercise, we're looking at a constant function, which is a special case with a very straightforward derivative, as you'll see below.

Constant Function

A constant function is one of the simplest types of functions in calculus. It takes the form \( f(x) = b \), where \( b \) is a constant value.

• No matter what input you use for \( x \), the output of the function remains the same.
• Graphically, this is represented as a horizontal line on the Cartesian plane.

The unchanging nature of a constant function gives an immediate clue that its derivative, or rate of change, should be zero. This is because there's no actual change in the function’s value as \( x \) varies.
As we proceed to find the derivative using the limit definition, this lack of change will become even more apparent.

Limit Definition

The limit definition of a derivative is a fundamental concept in calculus, providing the theoretical basis for understanding derivatives. For a function \( f(x) \), the derivative at any point \( x \) is defined as:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]

• The essence of this definition lies in predicting the behavior of functions as their inputs undergo an infinitesimally small change.
• It’s like zooming in endlessly on the curve of a function until it resembles a straight line, whose slope is the derivative.

In our specific case of a constant function \( f(x) = b \), substituting into this definition yields:\[ \lim_{{h \to 0}} \frac{{b - b}}{h} = \lim_{{h \to 0}} 0 = 0 \]
The derivative is zero, confirming that there's no change in the function, regardless of any tiny adjustments we make to \( x \). This is why the derivative of a constant function is always zero.