Problem 54 - Calculus Volume 1

Question

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=6 $$

Step-by-Step Solution

Answer

The derivative \(f^{ extprime}(x)\) is 0.

Step 1: Understand the function

First, recognize that the function given is a constant function, \(f(x) = 6\). This means no matter what value of \(x\) you choose, \(f(x)\) is always 6.

Step 2: Write the derivative definition

The derivative of a function \(f(x)\) at a point \(x\) is defined as \[ f^{ extprime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]

Step 3: Substitute the function into the formula

Since \(f(x) = 6\), we have \(f(x+h) = 6\) as well because it is a constant. Substitute these into the derivative definition:\[ f^{ extprime}(x) = \lim_{h \to 0} \frac{6 - 6}{h}. \]

Step 4: Simplify the expression

The numerator simplifies to \(0\), so the expression becomes:\[ f^{ extprime}(x) = \lim_{h \to 0} \frac{0}{h}. \]

Step 5: Solve the limit

Since \(\frac{0}{h} = 0\) for any \(h eq 0\), the limit simplifies to:\[ f^{ extprime}(x) = 0. \]

Step 6: Conclusion

The derivative of the constant function \(f(x) = 6\) is 0, which is expected because the slope of a constant function is always zero.

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is characterized by a rule that assigns exactly one value to every input, regardless of what the input is. For instance, the function given as an example, \(f(x) = 6\), does not change as \(x\) changes. This means:

• The output value (6 in this case) remains the same for every possible \(x\).
• Graphically, it is represented by a horizontal line parallel to the x-axis.

This consistent behavior—where the function value does not increase or decrease—is crucial to understanding why the derivative simplifies to zero. It is because the slope of a horizontal line is zero, indicating no rise or fall as you move along the line.

Derivative Definition

The derivative of a function is a central concept in calculus. It measures how a function changes as its input changes, offering insights into the rate of change or the slope of a function. The formal definition of a derivative is given by the limit:\[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\]This formula tells us:

• How a tiny change \(h\) in \(x\) affects the change in the function value.
• The resulting fraction before taking the limit represents the function's average rate of change over a small interval \(h\).
• As \(h\) approaches 0, this average rate of change converges to the instantaneous rate of change, defined as the derivative at \(x\).

By applying this definition to constant functions, we can understand why their derivatives are zero—since a constant function does not change, its rate of change is zero.

Limit Process

The limit process is a fundamental mathematical concept used in calculus and analysis. It allows us to precisely define and find values as an input approaches a certain point, even if the function is not defined exactly at that point. Applying the limit process to find a derivative involves:

• Taking the difference \(f(x+h) - f(x)\) and dividing by \(h\) to find the average rate of change of the function over the small interval \(h\).
• Letting \(h\) approach 0, which allows us to consider the behavior of the function at a specific point.
• For a constant function, this process highlights that the value of the function doesn't change even as \(h\) approaches 0, thus the numerator remains 0, leading to a derivative of 0.

This process is integral to understanding not just derivatives, but also a wide range of concepts in calculus, providing a deeper insight into how functions behave.