How To Differentiate X^6 Ln(7x) A Step-by-Step Guide

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Introduction to Differentiation

Differentiation, a fundamental concept in calculus, is the process of finding the derivative of a function. The derivative measures the instantaneous rate of change of a function, providing valuable insights into its behavior. In simpler terms, it tells us how much a function's output changes for a tiny change in its input. Understanding differentiation is crucial for various applications in mathematics, physics, engineering, economics, and computer science. It allows us to analyze the slopes of curves, find maximum and minimum values of functions, and model dynamic systems. The ability to differentiate complex functions is a core skill in advanced mathematical studies and practical problem-solving.

At its core, differentiation relies on the concept of limits. The derivative of a function at a specific point is defined as the limit of the difference quotient as the change in the input approaches zero. This limit represents the slope of the tangent line to the function's graph at that point. Mastering the techniques of differentiation enables us to move beyond simple linear functions and analyze the behavior of more complex functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions. The rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule, provide a systematic framework for finding derivatives of various types of functions. This article will delve into differentiating the function f(x) = x^6 ln(7x), providing a step-by-step explanation to ensure a thorough understanding of the process.

By applying the fundamental principles and rules of differentiation, we can effectively tackle intricate mathematical problems. Whether we are optimizing a business process, designing an engineering structure, or modeling a physical phenomenon, differentiation provides the tools to understand and manipulate dynamic systems. In the context of f(x) = x^6 ln(7x), we will explore how the combination of polynomial and logarithmic functions requires the application of both the product rule and the chain rule. The detailed walkthrough will not only demonstrate the mechanics of differentiation but also highlight the underlying mathematical reasoning. This approach ensures that readers gain not just the ability to compute derivatives but also a deeper conceptual understanding of calculus. The following sections will break down the differentiation process into manageable steps, making it accessible and comprehensible for learners of all levels.

Problem Statement: Differentiating f(x) = x^6 ln(7x)

To differentiate the function f(x) = x^6 ln(7x), we need to apply the fundamental principles of calculus, specifically the product rule and the chain rule. This function is a product of two simpler functions: x^6 and ln(7x). The product rule is essential when differentiating a function that is a product of two other functions, while the chain rule is necessary when dealing with composite functions, such as the natural logarithm of a function of x. The successful differentiation of f(x) depends on the correct application of these rules, along with a solid understanding of basic derivative formulas.

Understanding the problem requires us to recognize the structure of the function. f(x) = x^6 ln(7x) is not a simple function but rather a composite one. The term x^6 is a power function, and ln(7x) is a logarithmic function. The presence of 7x inside the logarithm necessitates the use of the chain rule, as it represents a function within a function. The product rule states that the derivative of two functions u(x) and v(x) is given by (uv)' = u'v + uv'. Identifying u(x) = x^6 and v(x) = ln(7x) sets the stage for applying this rule. Furthermore, differentiating ln(7x) involves the chain rule, which dictates that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this case, g(x) = 7x, and its derivative needs to be calculated and incorporated into the overall derivative of f(x).

Before diving into the step-by-step solution, it’s important to recap the necessary derivative formulas. The power rule states that the derivative of x^n is nx^(n-1). The derivative of the natural logarithm function ln(x) is 1/x. Combining these with the product and chain rules will allow us to systematically differentiate f(x). The following sections will break down the differentiation process, showing each step in detail. This approach ensures that readers can follow along and understand not only the mechanics but also the underlying concepts. By the end of this article, you will have a clear understanding of how to differentiate functions of this type and be well-equipped to tackle similar problems in calculus.

Applying the Product Rule

The product rule is a fundamental concept in calculus that provides a method for differentiating functions that are the product of two other functions. Mathematically, if we have a function f(x) = u(x)v(x), where u(x) and v(x) are differentiable functions, then the derivative f'(x) is given by:

f'(x) = u'(x)v(x) + u(x)v'(x)

This rule is essential for differentiating a wide range of functions, especially those that involve products of polynomials, trigonometric functions, exponential functions, and logarithmic functions. Understanding and applying the product rule correctly is crucial for solving many calculus problems, including the differentiation of f(x) = x^6 ln(7x). In this case, we can identify u(x) = x^6 and v(x) = ln(7x), which allows us to proceed with applying the rule.

The product rule's structure reflects the way products change. The derivative of the product is not simply the product of the derivatives; instead, it accounts for the individual rates of change of each function and their interaction. The term u'(x)v(x) represents the change in f(x) due to the change in u(x) while v(x) is held constant, and u(x)v'(x) represents the change in f(x) due to the change in v(x) while u(x) is held constant. Together, these terms provide the total rate of change of the product.

To apply the product rule to f(x) = x^6 ln(7x), we first need to identify the two functions being multiplied. Here, u(x) = x^6 and v(x) = ln(7x). Next, we need to find the derivatives of u(x) and v(x) separately. The derivative of u(x) = x^6 can be found using the power rule, which states that the derivative of x^n is nx^(n-1). Thus, u'(x) = 6x^(6-1) = 6x^5. The derivative of v(x) = ln(7x) requires the chain rule, which we will discuss in the next section. However, for the purpose of applying the product rule, we will denote v'(x) and calculate it explicitly later. With u(x), v(x), and u'(x) identified, we can now set up the product rule: f'(x) = u'(x)v(x) + u(x)v'(x) = 6x^5 ln(7x) + x^6 v'(x). The next step is to find v'(x) using the chain rule, which will complete the process of differentiating f(x).

Applying the Chain Rule to ln(7x)

The chain rule is another essential rule in calculus, particularly useful when differentiating composite functions. A composite function is a function that is composed of another function, meaning one function is nested inside another. For example, ln(7x) is a composite function because the logarithmic function ln(u) is composed with the function u(x) = 7x. The chain rule provides a method for differentiating such functions, stating that if y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). In simpler terms, we differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function. This rule is indispensable for handling more complex functions and is a cornerstone of differential calculus.

To understand the chain rule, it’s helpful to visualize it as a process of peeling away layers of a function. The outer function’s derivative is calculated first, while the inner function remains unchanged. Then, the derivative of the inner function is found and multiplied with the result from the outer function. This process ensures that we account for the rates of change at each level of composition. The chain rule is applicable not just to two-layer compositions but also to functions composed of multiple layers. The key is to work from the outermost layer inward, applying the rule sequentially.

In the context of differentiating v(x) = ln(7x), we can identify the outer function as f(u) = ln(u) and the inner function as g(x) = 7x. Applying the chain rule involves finding the derivatives of both functions. The derivative of the outer function, f'(u) = d/du [ln(u)], is 1/u. The derivative of the inner function, g'(x) = d/dx [7x], is 7. Now, according to the chain rule, v'(x) = f'(g(x)) * g'(x) = (1/g(x)) * g'(x) = (1/(7x)) * 7. Simplifying this expression gives v'(x) = 7/(7x) = 1/x. This result shows how the chain rule effectively handles the composition of functions, ensuring that all components of the rate of change are properly accounted for. With the derivative of ln(7x) found, we can now substitute this into the product rule formula to complete the differentiation of f(x) = x^6 ln(7x).

Combining the Results

After applying both the product rule and the chain rule, we can now combine the results to find the derivative of f(x) = x^6 ln(7x). We previously established that f'(x) = u'(x)v(x) + u(x)v'(x), where u(x) = x^6 and v(x) = ln(7x). We found that u'(x) = 6x^5 and, using the chain rule, v'(x) = 1/x. Substituting these into the product rule formula gives us:

f'(x) = (6x^5)ln(7x) + (x^6)(1/x)

This expression represents the complete derivative of f(x), but it can be simplified further to make it more concise and easier to work with. Simplification is an important step in calculus, as it often reveals underlying structure and facilitates further analysis or computation. In this case, we can simplify the second term by canceling out a factor of x.

To simplify the expression, we look for common factors or terms that can be combined. In the equation f'(x) = 6x^5 ln(7x) + x^6(1/x), the second term can be simplified by dividing x^6 by x, which yields x^5. Thus, the expression becomes:

f'(x) = 6x^5 ln(7x) + x^5

Now, we can see that both terms have a common factor of x^5. Factoring out x^5 from the entire expression gives us:

f'(x) = x^5 (6ln(7x) + 1)

This simplified form of the derivative is much cleaner and easier to interpret. It clearly shows the components of the derivative and their relationship. The simplified derivative, f'(x) = x^5 (6ln(7x) + 1), is the final answer, providing the rate of change of f(x) = x^6 ln(7x). This comprehensive solution demonstrates the effective application of the product and chain rules, combined with algebraic simplification techniques, to solve a typical calculus problem. The next section will summarize the steps and provide a recap of the key concepts.

Final Answer: f'(x) = x^5 (6ln(7x) + 1)

In summary, to differentiate the function f(x) = x^6 ln(7x), we followed a structured approach that involved applying the product rule and the chain rule. The final answer, after simplification, is f'(x) = x^5 (6ln(7x) + 1). This derivative represents the instantaneous rate of change of the function f(x) and provides valuable information about its behavior. The process highlighted the importance of understanding and correctly applying calculus rules, as well as the benefits of simplifying expressions for clarity and ease of use. This section will recap the key steps and concepts covered in the solution.

First, we recognized that f(x) is a product of two functions, u(x) = x^6 and v(x) = ln(7x), which necessitates the use of the product rule. The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). This rule is fundamental for differentiating functions that are formed by the product of two or more other functions. Correctly identifying the component functions and applying the rule accurately is crucial for solving such problems. Next, we found the derivatives of u(x) and v(x) separately. The derivative of u(x) = x^6 was found using the power rule, which states that the derivative of x^n is nx^(n-1). Thus, u'(x) = 6x^5. The derivative of v(x) = ln(7x) required the application of the chain rule. The chain rule is essential for differentiating composite functions, where one function is nested inside another. In this case, ln(7x) is a composition of the natural logarithm function and the linear function 7x.

The chain rule states that if y = f(g(x)), then y' = f'(g(x)) * g'(x). Applying this to v(x) = ln(7x), we identified the outer function as f(u) = ln(u) and the inner function as g(x) = 7x. The derivative of f(u) is 1/u, and the derivative of g(x) is 7. Therefore, v'(x) = (1/(7x)) * 7 = 1/x. Once we had the derivatives of u(x) and v(x), we substituted them into the product rule formula: f'(x) = (6x^5)ln(7x) + (x^6)(1/x). The final step was to simplify the expression. We simplified f'(x) = 6x^5 ln(7x) + x^6(1/x) by dividing x^6 by x, resulting in x^5. This gave us f'(x) = 6x^5 ln(7x) + x^5. We then factored out the common term x^5 to obtain the simplified derivative: f'(x) = x^5 (6ln(7x) + 1). This simplified form is the final answer and represents the derivative of the given function.

Practice Problems

To solidify your understanding of differentiation techniques, particularly the product and chain rules, working through practice problems is invaluable. These problems provide an opportunity to apply the concepts learned and reinforce the step-by-step processes involved. Here are a few additional problems that are similar in complexity and structure to the example we worked through:

  1. g(x) = x^4 sin(3x)
  2. h(x) = e^(2x) cos(x)
  3. k(x) = x^2 ln(5x)
  4. m(x) = (x^3 + 1)e^(-x)
  5. n(x) = √x * ln(x^2)

Each of these functions requires a combination of differentiation rules, including the product rule and chain rule, making them excellent practice for mastering these techniques. When solving these problems, remember to follow a systematic approach:

  • Identify the Structure: Determine if the function is a product, quotient, or composite function. This will guide you in selecting the appropriate differentiation rule (product rule, quotient rule, or chain rule).
  • Apply the Relevant Rule(s): If the function is a product, use the product rule. If it involves a composite function, use the chain rule. If it's a combination, apply the rules sequentially.
  • Find Derivatives of Individual Components: Differentiate each component function separately. For example, if using the product rule, find u'(x) and v'(x). If using the chain rule, find f'(g(x)) and g'(x).
  • Substitute and Simplify: Substitute the derivatives back into the appropriate rule and simplify the expression. Simplification often involves combining like terms, factoring, or canceling common factors.
  • Check Your Work: Review your steps to ensure no errors were made in applying the rules or simplifying the expression.

By working through these practice problems, you will develop confidence in your ability to differentiate complex functions. The key is to approach each problem methodically, breaking it down into manageable steps. With practice, you'll become more proficient at recognizing patterns and applying the correct differentiation techniques efficiently.

Conclusion

In conclusion, differentiating functions like f(x) = x^6 ln(7x) involves the methodical application of fundamental calculus rules such as the product rule and the chain rule. These rules, when combined with basic derivative formulas and algebraic simplification, allow us to find the rate of change of complex functions. The step-by-step approach outlined in this article provides a clear framework for tackling similar differentiation problems, emphasizing the importance of recognizing the structure of the function and applying the appropriate rules in the correct order.

This article has demonstrated how the product rule is essential for differentiating functions that are the product of two or more other functions. The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). Applying this rule requires identifying the component functions u(x) and v(x), finding their derivatives, and then substituting them into the formula. For the function f(x) = x^6 ln(7x), the components are u(x) = x^6 and v(x) = ln(7x), making the product rule the natural first step in differentiation. Furthermore, the chain rule plays a crucial role when dealing with composite functions. The chain rule states that if y = f(g(x)), then y' = f'(g(x)) * g'(x). This rule is necessary for differentiating functions where one function is nested inside another, such as ln(7x). In this case, the chain rule helps us find the derivative of ln(7x) by differentiating the outer function (ln(u)) and the inner function (7x) separately and then combining the results.

The process of simplifying the derivative is as important as applying the differentiation rules themselves. Simplification not only makes the expression more manageable but also can reveal underlying patterns and facilitate further analysis. In the example, we simplified the derivative f'(x) = 6x^5 ln(7x) + x^6(1/x) by canceling out common factors and combining like terms, ultimately arriving at f'(x) = x^5 (6ln(7x) + 1). This final form is much cleaner and easier to work with. By mastering these differentiation techniques and practicing with a variety of problems, readers can build a strong foundation in calculus and enhance their problem-solving skills. The ability to differentiate complex functions is a valuable asset in many fields, including mathematics, physics, engineering, and computer science. This article serves as a comprehensive guide to differentiating f(x) = x^6 ln(7x) and provides the tools and knowledge necessary to tackle similar challenges effectively.