Problem 27 - Precalculus: Mathematics for Calculus

Question

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=x^{2}+2 x$$

Step-by-Step Solution

Answer

The derivative \(f'(x) = 2x + 2\). \(f'(3) = 8\) and \(f'(4) = 10\).

Step 1: Define the Derivative Function

The derivative of a function describes its rate of change. For the function \(f(x) = x^2 + 2x\), we use the rules of differentiation to find \(f'(x)\).

Step 2: Apply Power Rule to Derivative

Differentiate the function term-by-term. The power rule states that for \(x^n\), the derivative is \(nx^{n-1}\). Thus, the derivative of \(x^2\) is \(2x\), and the derivative of \(2x\) is \(2\).

Step 3: Find the General Derivative

Combine the differentiated terms: \(f'(x) = 2x + 2\). This is the general form of the derivative for any value of \(a\) in the domain of \(f\).

Step 4: Evaluate the Derivative at Specific Points

To find \(f'(3)\), substitute \(x = 3\) into \(f'(x) = 2x + 2\). Similarly, substitute \(x = 4\) to find \(f'(4)\).

Step 5: Calculate \(f'(3)\)

Substitute \(x = 3\) into the derivative: \(f'(3) = 2(3) + 2 = 6 + 2 = 8\).

Step 6: Calculate \(f'(4)\)

Substitute \(x = 4\) into the derivative: \(f'(4) = 2(4) + 2 = 8 + 2 = 10\).

Key Concepts

Differentiation Rules

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative gives us insight into how the function behaves and changes. To differentiate a function, we apply a set of rules called the differentiation rules. These rules simplify the process of finding derivatives for various functions.

• Constant Rule: The derivative of a constant is zero. This reflects the fact that a constant function doesn't change as the variable changes.
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives. This rule allows us to handle each part of a function separately, before combining the results.
• Product Rule and Quotient Rule: These are more advanced rules used when functions are multiplied or divided. They provide a way to find derivatives without simplifying complex expressions into simpler terms.
• Chain Rule: This rule is crucial when composing functions. It tells us how to differentiate a function that is nested inside another function.

Mastering these rules is essential for anyone studying calculus, as they form the basis for more advanced topics and applications.

Power Rule

One of the most common differentiation techniques is the Power Rule. This rule is incredibly useful because it allows us to easily find the derivative of powers of variables. For any function in the form of \( x^n \), where \( n \) is a real number, the Power Rule states that its derivative is \( nx^{n-1} \).

Let's illustrate this with an example: To find the derivative of \( f(x) = x^2 \), apply the Power Rule by multiplying the exponent by the coefficient (which is 1 in this case), and then subtracting one from the exponent. This results in \( 2x^{1} \) or simply \( 2x \).

The Power Rule also applies to any polynomial function, where each term can be differentiated separately before being combined. This makes dealing with polynomials in calculus straightforward. Just remember to apply the rule to each term individually and then sum them up.

Rate of Change

In calculus, the derivative is often interpreted as the rate of change. This tells us how fast a function is changing at any given point. In the context of a graph, the derivative represents the slope of the tangent line to the function at a particular point.

When we talk about the rate of change, we're often interested in how one quantity changes in relation to another. For instance, if \( f(x) = x^2 + 2x \), then the derivative \( f'(x) = 2x + 2 \) gives us the rate at which the function \( f \) changes with respect to \( x \). By evaluating the derivative at specific points, like \( x = 3 \) or \( x = 4 \), we can find out exactly how fast the function is changing at these points, which in our example were 8 and 10 respectively.

The concept of rate of change is not just a foundational idea in calculus but also widely used in various fields such as physics, engineering, and economics to model and analyze dynamic systems.