Derivative Of X^2(2x+1)^4 A Step By Step Solution

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In the realm of calculus, finding the derivative of a function is a fundamental operation. It allows us to determine the instantaneous rate of change of a function, which has numerous applications in various fields such as physics, engineering, and economics. In this comprehensive guide, we will delve into the process of finding the derivative of the function f(x) = x2(2x+1)4. This function presents an interesting challenge as it involves the product of two terms, each of which is a composite function. To tackle this, we will employ a combination of the product rule and the chain rule, two essential tools in the calculus toolkit.

Understanding the Product Rule

The product rule is a cornerstone of differential calculus, providing a systematic way to find the derivative of a function that is expressed as the product of two other functions. This rule is indispensable when dealing with expressions like x2(2x+1)4, where we have two distinct components multiplied together. The essence of the product rule lies in recognizing that the rate of change of the entire product is influenced by the rates of change of each individual factor, as well as their current values. The rule elegantly captures this interplay, ensuring that we account for all contributions to the overall derivative. Mathematically, if we have two functions, u(x) and v(x), and we want to find the derivative of their product, u(x)v(x), the product rule states that:

(d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

In simpler terms, the derivative of the product is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. This might seem like a mouthful, but with practice, it becomes second nature. Applying the product rule correctly is crucial for many calculus problems, as it breaks down complex derivatives into more manageable parts. To illustrate its power, consider our function f(x) = x2(2x+1)4. We can identify u(x) = x^2 and v(x) = (2x+1)^4. By finding the derivatives of u(x) and v(x) separately, we can then use the product rule to find the derivative of the entire function. This approach allows us to systematically handle the complexities arising from the product of two functions, each with its own unique behavior. Mastering the product rule is not just about memorizing a formula; it's about understanding how the rates of change of individual factors combine to determine the rate of change of their product. This understanding is what makes the product rule such a powerful and versatile tool in calculus.

Applying the Chain Rule

The chain rule is another essential concept in calculus, particularly vital when dealing with composite functions. A composite function is essentially a function within a function, such as (2x+1)^4 in our example. The chain rule provides a method for differentiating these nested functions, allowing us to break down complex derivatives into simpler steps. The core idea behind the chain rule is to account for how the inner function's change affects the outer function's change. It acknowledges that the rate of change of the composite function depends on both the rate of change of the outer function and the rate of change of the inner function. Mathematically, if we have a composite function y = f(g(x)), where g(x) is the inner function and f(x) is the outer function, the chain rule states that:

dy/dx = f'(g(x)) * g'(x)

In simpler terms, the derivative of the composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This may sound complicated, but it’s a systematic way to handle functions that are built upon other functions. To illustrate, let's consider v(x) = (2x+1)^4 from our original function. Here, the outer function is f(u) = u^4 and the inner function is g(x) = 2x+1. Applying the chain rule, we first find the derivative of the outer function, which is f'(u) = 4u^3. Then, we evaluate this at the inner function, giving us 4(2x+1)^3. Next, we find the derivative of the inner function, which is g'(x) = 2. Finally, we multiply these two results together, giving us the derivative of v(x) as 4(2x+1)^3 * 2 = 8(2x+1)^3. The chain rule is not just a formula to memorize; it's a method of unraveling the layers of a composite function to find its derivative. By understanding how each layer contributes to the overall rate of change, we can confidently differentiate even the most complex composite functions. Mastering the chain rule is crucial for success in calculus, as it appears in a wide range of problems and is a fundamental tool for advanced mathematical concepts.

Step-by-Step Solution for Finding the Derivative of x2(2x+1)4

Let's now apply the product rule and chain rule to find the derivative of f(x) = x2(2x+1)4. This step-by-step approach will not only provide the solution but also solidify the understanding of the rules involved. The key to solving complex calculus problems is to break them down into manageable parts, and this example is no exception. We'll start by identifying the components of the function and then apply the appropriate rules systematically.

  1. Identify u(x) and v(x):

    We begin by recognizing that f(x) is a product of two functions: u(x) = x^2 and v(x) = (2x+1)^4. This is the first step in applying the product rule. Identifying these components correctly is crucial, as it sets the stage for the subsequent steps. With u(x) and v(x) clearly defined, we can now focus on finding their individual derivatives.

  2. Find u'(x):

    The derivative of u(x) = x^2 is straightforward using the power rule, which states that the derivative of x^n is nx^(n-1). Applying this rule, we get u'(x) = 2x. This derivative represents the rate of change of x^2 with respect to x. It's a simple yet essential component in the product rule, and accurately finding u'(x) is key to the correct final answer.

  3. Find v'(x):

    Finding the derivative of v(x) = (2x+1)^4 requires the chain rule, as it is a composite function. As discussed earlier, the chain rule involves differentiating the outer function and the inner function separately and then combining them. The outer function here is u^4 and the inner function is 2x+1. First, we differentiate the outer function with respect to u, which gives us 4u^3. Then, we substitute the inner function back in, resulting in 4(2x+1)^3. Next, we differentiate the inner function 2x+1 with respect to x, which gives us 2. Finally, we multiply these results together, giving us v'(x) = 4(2x+1)^3 * 2 = 8(2x+1)^3. This derivative represents the rate of change of (2x+1)^4 with respect to x, and it's a crucial component in applying the product rule.

  4. Apply the Product Rule:

    Now that we have u(x), v(x), u'(x), and v'(x), we can apply the product rule: (d/dx)[u(x)v(x)] = u'(x)v(x) + u(x)v'(x). Substituting the functions and their derivatives, we get:

    f'(x) = (2x)(2x+1)^4 + (x2)(8(2x+1)3)

    This expression represents the derivative of the original function f(x). It combines the rates of change of u(x) and v(x), weighted by their current values, as dictated by the product rule. While this is technically the derivative, it's often beneficial to simplify it further for ease of use and interpretation.

  5. Simplify the Expression:

    To simplify the derivative, we can factor out common terms. Notice that both terms in the expression have factors of (2x+1)^3 and 2x. Factoring these out, we get:

    f'(x) = 2x(2x+1)^3[(2x+1) + 4x]

    Further simplifying the expression inside the brackets:

    f'(x) = 2x(2x+1)^3(2x + 1 + 4x)

    f'(x) = 2x(2x+1)^3(6x + 1)

    This simplified expression is the final derivative of the function f(x) = x2(2x+1)4. It's more compact and easier to work with than the initial derivative we obtained from the product rule. Simplifying the derivative is not just about making it look nicer; it often makes it easier to analyze the function's behavior, such as finding critical points or determining intervals of increase and decrease. This step completes the process of finding the derivative, showcasing the power of the product rule and chain rule in handling complex functions.

Conclusion

In conclusion, finding the derivative of f(x) = x2(2x+1)4 demonstrates the application of two fundamental calculus rules: the product rule and the chain rule. By systematically breaking down the function into its components, applying the appropriate rules, and simplifying the result, we arrive at the derivative: f'(x) = 2x(2x+1)^3(6x + 1). This process underscores the importance of understanding and mastering these calculus techniques for solving more complex problems in mathematics and related fields. Mastering these rules is not just about finding derivatives; it's about developing a deeper understanding of how functions change and interact, which is a cornerstone of calculus and its applications. The journey from the initial function to its derivative highlights the elegance and power of calculus in unraveling the behavior of complex mathematical expressions.

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