Differentiating G(x) = (3x³ + 2x)⁶ A Step-by-Step Guide

#mainkeyword Differentiating composite functions like g(x)=(3x³+2x)⁶ is a fundamental concept in calculus. This article provides a comprehensive guide on how to differentiate this function, focusing on the application of the chain rule. The chain rule is a powerful tool that allows us to find the derivative of a composite function, which is a function within a function. In this particular case, we have an outer function raised to the power of 6 and an inner function of a polynomial. Mastering this technique is crucial for various applications in mathematics, physics, engineering, and other scientific fields. Understanding the chain rule not only helps in differentiating complex functions but also builds a solid foundation for more advanced calculus concepts. This article aims to break down the process step-by-step, making it accessible to students and anyone interested in calculus. We'll begin by revisiting the basic rules of differentiation, then delve into the chain rule itself, and finally apply it to the given function. We'll also explore common pitfalls and provide tips to avoid them. By the end of this guide, you'll be equipped with the knowledge and skills to confidently differentiate similar composite functions. Whether you're a student preparing for an exam, a professional needing a refresher, or simply someone curious about calculus, this article will serve as a valuable resource. So, let's embark on this journey of differentiation and unravel the intricacies of the chain rule together.

Understanding the Chain Rule

At its core, the chain rule is a formula for finding the derivative of a composite function. A composite function is essentially a function that is composed of another function, meaning one function is nested inside another. To effectively apply the chain rule, it's essential to first identify the outer and inner functions within the composite function. 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, this can be represented as: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This formula might seem daunting at first, but with practice and a clear understanding of its components, it becomes a powerful tool. Let's break down the components of the chain rule formula. f(g(x)) represents the composite function, where g(x) is the inner function and f is the outer function. f'(g(x)) represents the derivative of the outer function f, evaluated at the inner function g(x). g'(x) represents the derivative of the inner function g(x). The chain rule is applicable in numerous situations where functions are nested within each other. For example, trigonometric functions of polynomials, exponential functions of other functions, and, as in our case, polynomial functions raised to a power. Recognizing when to apply the chain rule is a crucial skill in calculus. The presence of nested functions is a clear indicator that the chain rule is needed. Failing to recognize this and attempting to differentiate the function using simpler rules can lead to incorrect results. To illustrate, consider the function sin(x²). Here, the outer function is sin(u) and the inner function is u = x². The chain rule tells us that the derivative of sin(x²) is cos(x²) * 2x. In the next section, we will apply the chain rule specifically to our example function, g(x) = (3x³ + 2x)⁶, demonstrating the practical application of this fundamental rule.

Breaking Down g(x) = (3x³ + 2x)⁶

To effectively differentiate g(x) = (3x³ + 2x)⁶, we must first identify the outer and inner functions. This is a crucial step in applying the chain rule. In this case, the outer function, let's call it f(u), is the function raised to the power of 6, so f(u) = u⁶. The inner function, g(x), is the polynomial expression inside the parentheses, which is g(x) = 3x³ + 2x. Now that we've identified the outer and inner functions, we can proceed with finding their derivatives. This is a necessary step before we can apply the chain rule formula. Let's start with the outer function, f(u) = u⁶. To find its derivative, f'(u), we use the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹. Applying the power rule to f(u) = u⁶, we get f'(u) = 6u⁵. Remember, we're finding the derivative with respect to u here, as u is the variable in the outer function. Next, we need to find the derivative of the inner function, g(x) = 3x³ + 2x. To find g'(x), we'll again use the power rule, along with the sum/difference rule and the constant multiple rule. The power rule, as mentioned before, states that the derivative of xⁿ is nxⁿ⁻¹. The sum/difference rule states that the derivative of a sum or difference of terms is the sum or difference of their derivatives. The constant multiple rule states that the derivative of a constant times a function is the constant times the derivative of the function. Applying these rules to g(x) = 3x³ + 2x, we get: The derivative of 3x³ is 3 * 3x² = 9x². The derivative of 2x is 2. Therefore, g'(x) = 9x² + 2. We now have the derivatives of both the outer and inner functions. f'(u) = 6u⁵ and g'(x) = 9x² + 2. The next step is to apply the chain rule formula, which we'll do in the subsequent section. By breaking down the composite function into its components and finding their individual derivatives, we've set the stage for a successful application of the chain rule.

Applying the Chain Rule: Step-by-Step

With the derivatives of the outer and inner functions calculated, we can now apply the chain rule to differentiate g(x) = (3x³ + 2x)⁶. The chain rule, as we recall, states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). We've already determined that f'(u) = 6u⁵ and g'(x) = 9x² + 2. The next step is to substitute the inner function, g(x), into the derivative of the outer function, f'(u). This means replacing u in f'(u) = 6u⁵ with 3x³ + 2x. So, f'(g(x)) = 6(3x³ + 2x)⁵. Now, we have all the components needed to apply the chain rule formula. We have f'(g(x)) = 6(3x³ + 2x)⁵ and g'(x) = 9x² + 2. Multiplying these together gives us the derivative of g(x). Therefore, g'(x) = f'(g(x)) * g'(x) = 6(3x³ + 2x)⁵ * (9x² + 2). This is the derivative of the original function, g(x) = (3x³ + 2x)⁶. Let's recap the steps we took to differentiate this function using the chain rule: First, we identified the outer and inner functions. Then, we found the derivatives of both the outer and inner functions. Next, we substituted the inner function into the derivative of the outer function. Finally, we multiplied the result by the derivative of the inner function. This step-by-step process ensures a clear and accurate application of the chain rule. While we have found the derivative, it is often beneficial to simplify the expression if possible. However, in this case, further simplification might not be necessary or might make the expression more complex. The current form, g'(x) = 6(3x³ + 2x)⁵ * (9x² + 2), clearly shows the application of the chain rule and the components involved. In the following section, we will explore some common mistakes to avoid when using the chain rule.

Common Mistakes to Avoid

When applying the chain rule, several common mistakes can lead to incorrect results. Recognizing and avoiding these pitfalls is crucial for mastering differentiation. One of the most frequent errors is forgetting to multiply by the derivative of the inner function. The chain rule explicitly states that 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. Omitting this multiplication is a significant error that invalidates the entire process. For instance, in our example, if we only calculated 6(3x³ + 2x)⁵ and neglected to multiply by (9x² + 2), we would have obtained an incorrect derivative. Another common mistake is incorrectly identifying the outer and inner functions. A clear understanding of function composition is essential for correctly applying the chain rule. If the outer and inner functions are misidentified, the subsequent steps will be based on a flawed foundation, leading to an incorrect derivative. For example, if we were to mistakenly consider (3x³ + 2x) as the outer function and x⁶ as the inner function, our differentiation would be completely off track. A third common error is applying the power rule incorrectly, especially when dealing with complex inner functions. The power rule, which states that the derivative of xⁿ is nxⁿ⁻¹, is often used in conjunction with the chain rule. However, it's crucial to apply it correctly to both the outer and inner functions. Errors in applying the power rule can arise from incorrect exponents or coefficients. A fourth mistake involves errors in algebraic manipulation. Differentiation often involves simplifying the resulting expression after applying the rules of calculus. Mistakes in algebra, such as incorrect distribution or combining like terms, can lead to a correct derivative being simplified into an incorrect form. Therefore, attention to detail in algebraic manipulation is as important as understanding the calculus concepts. Finally, a lack of practice can contribute to errors. The chain rule, like any mathematical concept, requires practice to master. Working through various examples and problems helps solidify understanding and build confidence in applying the rule correctly. By being aware of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and proficiency in differentiating composite functions using the chain rule. In the next section, we will summarize the key steps and concepts covered in this guide.

Conclusion: Mastering Differentiation with the Chain Rule

In this guide, we've explored the process of differentiating the function g(x) = (3x³ + 2x)⁶ using the chain rule. We've broken down the concept into manageable steps, starting with an understanding of the chain rule itself, then identifying the outer and inner functions, finding their derivatives, applying the chain rule formula, and finally, discussing common mistakes to avoid. The chain rule is a fundamental concept in calculus that enables us to differentiate composite functions, which are functions nested within other functions. It 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. Mastering this rule is crucial for tackling a wide range of differentiation problems. We began by identifying the outer function, f(u) = u⁶, and the inner function, g(x) = 3x³ + 2x, in our example function. Then, we found their respective derivatives: f'(u) = 6u⁵ and g'(x) = 9x² + 2. We then applied the chain rule formula, substituting the inner function into the derivative of the outer function and multiplying by the derivative of the inner function. This yielded the derivative g'(x) = 6(3x³ + 2x)⁵ * (9x² + 2). Throughout this process, we emphasized the importance of each step and the reasoning behind it. We also highlighted common mistakes to avoid, such as forgetting to multiply by the derivative of the inner function, misidentifying the outer and inner functions, applying the power rule incorrectly, making algebraic errors, and lacking sufficient practice. By understanding these pitfalls, you can minimize errors and improve your accuracy in differentiation. In conclusion, differentiating composite functions using the chain rule requires a systematic approach and a clear understanding of the underlying concepts. By practicing the steps outlined in this guide and being mindful of common mistakes, you can develop the skills and confidence to tackle even more complex differentiation problems. The chain rule is a powerful tool that extends far beyond this specific example, and its mastery will serve you well in various areas of mathematics, science, and engineering.