Finding The Derivative: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of derivatives, specifically tackling the question: What is the first derivative of the function f(x) = (2x+8)(x-2)? Don't worry, it's not as scary as it sounds. We'll break it down step by step, making sure you grasp the concept and can confidently solve similar problems. Ready? Let's get started!

Understanding Derivatives: The Basics

Alright, before we jump into the nitty-gritty, let's quickly recap what a derivative actually is. In simple terms, the derivative of a function tells us the rate at which the function's output changes with respect to its input. Think of it as the slope of the tangent line at any given point on the function's curve. It’s like zooming in on a specific spot on the graph and finding the steepness there. The derivative gives us that information. So, when we talk about f'(x), we're referring to the derivative of the function f(x). It represents how f(x) changes. Remember, this concept is fundamental to calculus and has applications in various fields like physics, engineering, and economics.

Now, there are several ways to find a derivative. For this problem, we can use a couple of different approaches, and we'll go through both to make sure everything clicks. The main goal here is to help you understand the process and build your confidence in solving derivative problems. We will explore the product rule (though we can avoid it with some algebraic manipulation) and the power rule. The power rule is a key tool for differentiating polynomials. It states that if you have a term like x raised to a power (e.g., x², x³), its derivative is the power multiplied by x raised to the power minus one. For instance, the derivative of x² is 2x, and the derivative of x³ is 3x². This rule simplifies a lot of calculations. Also, don't forget the constant multiple rule. If a constant is multiplying a function, you can simply multiply the derivative of the function by that constant. These rules, when combined, make it easier to solve problems involving derivatives, which are crucial for understanding rates of change and optimization in various applications. Let's start with a straightforward method.

Method 1: Expanding and Differentiating

This method is super intuitive and is often the best for beginners. Here's how it works:

  1. Expand the Function: First, let's expand the original function f(x) = (2x + 8)(x - 2). We do this by multiplying the terms: f(x) = 2x * x - 2x * 2 + 8 * x - 8 * 2 f(x) = 2x² - 4x + 8x - 16 f(x) = 2x² + 4x - 16

    So now, f(x) = 2x² + 4x - 16.

  2. Differentiate Term by Term: Now, let’s find the derivative of each term separately.

    • The derivative of 2x² is 4x (using the power rule: 2 * 2 * x^(2-1) = 4x).
    • The derivative of 4x is 4 (the power rule: 1 * 4 * x^(1-1) = 4 * x⁰ = 4 * 1 = 4).
    • The derivative of -16 is 0 (the derivative of a constant is always zero).

  3. Combine the Derivatives: Add the derivatives of each term together.

    f'(x) = 4x + 4 - 0 f'(x) = 4x + 4

    And there you have it! The derivative of f(x) = (2x + 8)(x - 2) is f'(x) = 4x + 4. Easy peasy, right? This approach is generally easier to follow for many, as it breaks down the problem into smaller, more manageable parts. By expanding first, you avoid having to apply the product rule directly, making the calculation more straightforward and less prone to errors. This is especially helpful if you're just starting out with derivatives. Understanding this method lays a strong foundation for tackling more complex derivative problems in the future. The ability to manipulate and simplify expressions before differentiating is a valuable skill in calculus. Therefore, practicing this method will improve your speed and accuracy in solving problems. It’s also a good way to double-check your work, particularly when dealing with the product rule. Always simplify and reduce the equation before applying any calculus rules, if possible!

Method 2: Using the Product Rule (Just for Reference)

Although we already solved the problem, let's talk about the product rule for informational purposes. This is super helpful when you have a function that’s the product of two other functions. The product rule states:

If f(x) = u(x) * v(x), then f'(x) = u'(x) * v(x) + u(x) * v'(x).

In our case, u(x) = 2x + 8 and v(x) = x - 2.

  1. Find the Derivatives of u(x) and v(x):

    • u'(x) = 2 (the derivative of 2x + 8).
    • v'(x) = 1 (the derivative of x - 2).

  2. Apply the Product Rule: Now, we plug these values into the product rule formula.

    f'(x) = 2 * (x - 2) + (2x + 8) * 1 f'(x) = 2x - 4 + 2x + 8 f'(x) = 4x + 4

    As you can see, we arrive at the same answer: f'(x) = 4x + 4. The product rule can be handy, but sometimes, as we've shown, it's easier to expand the function first, especially when the expansion is straightforward. This is especially true when dealing with polynomials. When working with complex functions, using the product rule directly may be necessary. By knowing both methods, you can choose the one that's most efficient for a given problem.

Choosing the Right Method

So, which method should you use? Well, it depends on the function and your comfort level. If expanding the function is simple, like in our case, it's often the quickest way to find the derivative. This avoids the extra steps of the product rule. However, it's essential to understand the product rule. As functions become more complex, expanding might not be an option, and you'll need the product rule. Remember that practice is key, and as you solve more problems, you'll become better at recognizing which method is the most efficient. This is the beauty of mathematics: there's often more than one way to solve a problem! Choosing the right method improves your efficiency in solving the problem. So, go ahead, and choose the most suitable method for you.

Summary and Conclusion

Alright, guys, let's recap! We’ve successfully found the derivative of f(x) = (2x + 8)(x - 2). The answer is f'(x) = 4x + 4. We walked through two methods: expanding and differentiating, and using the product rule. Remember, practice is key. The more problems you solve, the more comfortable you'll become with derivatives. Keep in mind the derivative represents the instantaneous rate of change. Understanding this concept is fundamental to calculus and essential for solving various real-world problems. Keep practicing and keep learning! You've got this!

So the correct answer is (D) f'(x) = 4x + 4!