Derivative Of F(x) = (4x + 1)^2: Step-by-Step Solution

by TextBrain Team 55 views

Hey guys! Today, we're diving into a common calculus problem: finding the derivative of a composite function. Specifically, we'll tackle the function F(x) = (4x + 1)^2. This might seem intimidating at first, but don't worry! We'll break it down step-by-step, so you'll be a pro in no time. Understanding derivatives is super crucial in calculus because it helps us understand how a function changes. This has applications in all sorts of fields, from physics and engineering to economics and computer science. So, let's get started and unlock the secrets of this derivative!

Understanding Derivatives

Before we jump into the specific problem, let's quickly recap what a derivative actually is. The derivative of a function at a certain point represents the instantaneous rate of change of the function at that point. Think of it as the slope of the line tangent to the function's graph at that point. This concept is fundamental in calculus and has wide-ranging applications.

Why are derivatives so important? Well, they help us understand the behavior of functions. For example, the derivative can tell us where a function is increasing or decreasing, where it reaches its maximum or minimum values, and the concavity of its graph. These insights are invaluable in optimization problems, curve sketching, and many other areas.

There are several ways to find derivatives. For simple functions, we can use the power rule, the constant multiple rule, and other basic rules. However, for more complex functions like F(x) = (4x + 1)^2, we need to employ the chain rule, which we'll discuss in detail later. Grasping the basics first will make the more complex rules much easier to handle. Make sure you're comfortable with basic algebraic manipulations as well, as they often come into play when simplifying derivatives.

Methods to Calculate the Derivative of F(x) = (4x + 1)^2

There are primarily two methods we can use to find the derivative of F(x) = (4x + 1)^2. We can either expand the function first and then apply the power rule, or we can directly apply the chain rule. Both methods are valid and will lead to the same answer, but one might be more efficient depending on your preference and the complexity of the function.

Method 1: Expanding and Applying the Power Rule

This method involves first expanding the squared term and then differentiating the resulting polynomial. Let's break it down:

  1. Expand the function: F(x) = (4x + 1)^2 = (4x + 1)(4x + 1) = 16x^2 + 8x + 1
  2. Apply the power rule: The power rule states that the derivative of x^n is nx^(n-1). We'll apply this rule to each term in the expanded function.

Now, let's differentiate each term:

  • The derivative of 16x^2 is 16 * 2x^(2-1) = 32x
  • The derivative of 8x is 8 * 1x^(1-1) = 8
  • The derivative of 1 (a constant) is 0
  1. Combine the results: Therefore, the derivative of F(x) = 16x^2 + 8x + 1 is F'(x) = 32x + 8.

This method is straightforward and relies on familiar rules of differentiation. It's a good choice when the function can be easily expanded.

Method 2: Applying the Chain Rule

The chain rule is a fundamental rule in calculus used to differentiate composite functions. A composite function is a function that is composed of two or more functions, like our example F(x) = (4x + 1)^2. In this case, we have an "outer" function (squaring) and an "inner" function (4x + 1). The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Mathematically, if we have a composite function F(x) = g(h(x)), then the chain rule tells us that F'(x) = g'(h(x)) * h'(x).

Let's apply the chain rule to F(x) = (4x + 1)^2:

  1. Identify the outer and inner functions:
    • Outer function: g(u) = u^2
    • Inner function: h(x) = 4x + 1
  2. Find the derivatives of the outer and inner functions:
    • g'(u) = 2u
    • h'(x) = 4
  3. Apply the chain rule: F'(x) = g'(h(x)) * h'(x) = 2(4x + 1) * 4
  4. Simplify: F'(x) = 8(4x + 1) = 32x + 8

As you can see, we arrive at the same result as with the expansion method. The chain rule might seem a bit more abstract at first, but it's incredibly powerful for differentiating complex functions.

Step-by-Step Solution Using the Chain Rule (Detailed)

Let's walk through the chain rule method again, but this time with even more detail, to make sure everyone's on board.

  1. Write down the function: F(x) = (4x + 1)^2
  2. Identify the outer function: The outer function is the squaring operation. We can think of it as something being raised to the power of 2. Let's call it g(u) = u^2, where 'u' represents whatever is inside the parentheses.
  3. Identify the inner function: The inner function is what's inside the parentheses, which is 4x + 1. Let's call it h(x) = 4x + 1.
  4. Find the derivative of the outer function: g'(u) = 2u (using the power rule).
  5. Find the derivative of the inner function: h'(x) = 4 (the derivative of 4x is 4, and the derivative of 1 is 0).
  6. Apply the chain rule formula: F'(x) = g'(h(x)) * h'(x). This means we need to plug the inner function (4x + 1) into the derivative of the outer function (2u), and then multiply the result by the derivative of the inner function (4).
  7. Substitute and simplify:
    • g'(h(x)) = 2(4x + 1)
    • F'(x) = 2(4x + 1) * 4
    • F'(x) = 8(4x + 1)
    • F'(x) = 32x + 8

And there you have it! We've successfully found the derivative of F(x) = (4x + 1)^2 using the chain rule. Breaking it down into these steps can make the process much clearer.

Common Mistakes to Avoid

When calculating derivatives, especially using the chain rule, there are a few common pitfalls to watch out for. Being aware of these can save you a lot of headaches!

  • Forgetting the chain rule: One of the biggest mistakes is forgetting to multiply by the derivative of the inner function. Remember, the chain rule is essential for composite functions.
  • Incorrectly identifying the inner and outer functions: It's crucial to correctly identify which part of the function is the "outer" function and which is the "inner" function. Practice can help with this.
  • Making algebraic errors: Derivatives often involve algebraic simplification. Be careful with your algebra to avoid mistakes in the final answer.
  • Not simplifying the final answer: While not strictly an error, it's good practice to simplify your derivative as much as possible. This makes it easier to work with in subsequent calculations.

By keeping these common mistakes in mind, you'll be well on your way to mastering derivatives!

Practice Problems

To really solidify your understanding, let's tackle a few practice problems. The best way to learn calculus is by doing it, so grab a pencil and paper and give these a try!

  1. Find the derivative of G(x) = (3x - 2)^3
  2. Find the derivative of H(x) = sin(2x)
  3. Find the derivative of J(x) = e(x2)

Try solving these using the chain rule and the power rule (where applicable). Don't be afraid to make mistakes – that's how we learn! If you get stuck, review the steps we discussed earlier and try breaking down the problem into smaller parts.

Conclusion

So, guys, we've successfully navigated the derivative of F(x) = (4x + 1)^2! We explored two methods: expanding and using the power rule, and directly applying the chain rule. We also highlighted common mistakes to avoid and provided some practice problems to help you master this concept. Remember, the chain rule is a powerful tool for differentiating composite functions, and understanding it is key to success in calculus.

Keep practicing, and don't hesitate to review the steps and explanations whenever you need a refresher. Happy calculating!