Problem 49 - Contemporary Precalculus

Question

Find the rule of the derivative of the function \(f .\) [See Example 7 and the remarks following it.] $$f(x)=x^{2}+x$$

Step-by-Step Solution

Answer

Answer: The derivative of the function \(f(x) = x^2 + x\) is \(f'(x) = 2x + 1\).

Step 1: Identify the terms of the function

The given function \(f(x)\) has two terms: \(x^2\) and \(x\). We will differentiate each term separately and then add them together to find the derivative of the entire function.

Step 2: Apply the power rule to the first term

The power rule states that if \(f(x) = x^n\), where \(n\) is a constant, then \(f'(x) = nx^{n-1}\). In our case, the first term is \(x^2\) (\(n = 2\)). Applying the power rule, we get: $$\frac{d}{dx}(x^2) = 2x^{2-1} = 2x$$

Step 3: Apply the power rule to the second term

Now, we will differentiate the second term of the function, which is \(x\). Since \(x\) is equal to \(x^1\) (\(n=1\)), we will apply the power rule again: $$\frac{d}{dx}(x) = 1x^{1-1} = 1x^0 = 1$$

Step 4: Add the derivatives of the terms

Next, we will add the derivatives calculated in the previous two steps: $$f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(x) = 2x + 1$$

Step 5: Write the final answer

The rule of the derivative of the function \(f(x) = x^2 + x\) is: $$f'(x) = 2x + 1$$

Key Concepts

Derivative Rules

The derivative rules are essential tools used to find the rate of change or the "slope" of a function. These rules simplify the process of differentiation by providing formulas and techniques that apply to common types of functions.
A derivative gives us an idea of how a function behaves, mapping its changes.

• The Sum Rule allows us to differentiate functions that are sums of other functions. So, for the function \(f(x) = x^2 + x\), we differentiate each term separately.
• The Constant Rule states that if you differentiate a constant, the result is zero.
• The General Differentiation Rule provides a pathway to apply derivatives when dealing with complex functions, like sums, products, or compositions.

By understanding and applying these rules, especially together, we can efficiently find the derivative of various functions, which is crucial in calculus.

Power Rule

The Power Rule is one of the most straightforward and widely used differentiation techniques. It simplifies finding the derivative of power functions, which are functions in the form of \(x^n\), where \(n\) is any real number.
To use the Power Rule:

• Identify the exponent \(n\) in the term \(x^n\).
• Differentiate by multiplying the entire term by \(n\) and then subtracting one from the exponent.

For instance, with the term \(x^2\) in our exercise, we apply the Power Rule: \(\frac{d}{dx}(x^2) = 2x^{2-1} = 2x\).
This method is efficient, helping us tackle polynomial functions without the hassle of additional methods. Recognizing the Power Rule ensures you are ready to deal with any polynomial term in calculus.

Differentiation Techniques

Differentiation techniques, such as the Power Rule and others, help us break down complex functions into manageable parts. Once identified, we can apply these techniques to solve multiple terms individually.
In the given exercise, we executed:

• The Power Rule separately for each term in the function \(f(x) = x^2 + x\).
• Added the derivatives of these individual terms to arrive at the overall derivative.

Understanding different methods, like applying the Sum Rule or tackling each function's component separately, provides a systematic approach to finding derivatives.
This ensures we not only find the solution but deeply comprehend how parts of a function contribute to its overall rate of change.

Function Analysis

Function analysis involves studying a function's properties, including its graph, increase or decrease intervals, and concavity. Understanding a function's derivative is crucial for this analysis, as it informs about the slope at any given point.

• The derivative of a function, like \(f'(x) = 2x + 1\), tells us the slope at any \(x\).
• Positive derivatives indicate increasing behavior, while negative derivatives suggest decreasing behavior.

Knowing the behavior and structure of the derivative gives insights into the original function, \(f(x) = x^2 + x\).
This enables us to predict and plot the function accurately, vital for fields dependent on precise mathematical modeling, such as physics and economics.