Problem 29 - Precalculus with Limits

Question

In Exercises 29-42, find the derivative of the function. \(f(x) = 5\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 5\) is 0.

Step 1: Identify the type of function

The given function \(f(x) = 5\) is a constant function. It's a function that always returns the same value no matter what the input is.

Step 2: Apply the differentiation rule

For any constant \(a\), the derivative is zero. This property comes directly from the definition of the derivative as the limit of the rate of change of the function, and since a constant function does not change, its rate of change is zero. Therefore, we apply this property to our function.

Step 3: Write down the derivative

By applying the rule from Step 2, the derivative of the function \(\)f(x) = 5\(\) is 0.

Key Concepts

Constant Function

When we talk about a constant function, we're referring to a very simple concept in mathematics. A constant function is one that returns the same value no matter the input. For example, the function \(f(x) = 5\) always outputs 5, regardless of what \(x\) is. This is why it's called a "constant" function.

Constant functions are straightforward to understand because they don't involve any variables other than the constant itself. They play a crucial role in calculus as the simplest members of the family of functions. To recognize a constant function, remember:

• The output is always the same, no matter the input.
• It's represented graphically as a horizontal line on the Cartesian plane.

Understanding constant functions is key to mastering differentiation, as they provide an excellent starting point for learning about derivative concepts.

Differentiation Rules

Differentiation is a core concept in calculus that deals with calculating the derivative of a function. The derivative represents an instantaneous rate of change and is a fundamental tool for understanding how functions behave.

The differentiation rules provide a set of guidelines to find derivatives efficiently. For a constant function like \(f(x) = 5\), the differentiation rule related to constants is very straightforward: the derivative of any constant function is always zero. This is because the slope of the horizontal line (graph of a constant function) is zero, meaning no change occurs.

Some key points to remember about differentiation rules include:

• The derivative of a constant is always zero.
• Basic rules also include the power rule, product rule, quotient rule, and chain rule, which are used for more complex functions.

Knowing these rules by heart helps you tackle derivative problems with confidence and ease.

Rate of Change

The rate of change is a fundamental concept that measures how a quantity changes in relation to another. In calculus, the derivative of a function provides this rate of change.

Let's consider the constant function \(f(x) = 5\). Because this function never changes regardless of \(x\), its rate of change is zero. This is why the derivative of a constant function, as determined by differentiation, is zero.

Think of rate of change as the speedometer of a function—it tells you how quickly values are shifting:

• For a constant function, the speed is zero because it represents no movement (change).
• Understanding the rate of change helps in various real-life applications, like computing speed, growth rates, and even economic trends.

The concept of rate of change is pivotal for interpreting real-world problems through the lens of mathematics, enabling you to predict and understand behaviors over time.