Differentiating E^(3x) / X^6 A Step-by-Step Calculus Guide
Introduction
In the realm of calculus, differentiating functions is a fundamental operation that allows us to determine the rate of change of a function with respect to its input variable. In this comprehensive guide, we will delve into the process of differentiating the function F(x) = e^(3x) / x^6. This function involves both an exponential term (e^(3x)) and a power term (x^6) in the denominator, making it an excellent example to illustrate the application of the quotient rule and chain rule in differentiation. Mastering such differentiations is crucial for solving a wide range of problems in physics, engineering, economics, and other quantitative disciplines. The ability to accurately differentiate complex functions like F(x) enables us to model and analyze real-world phenomena, optimize systems, and make informed predictions. This article will provide a step-by-step approach, ensuring a clear understanding of each stage in the differentiation process, making it a valuable resource for students, educators, and professionals alike. This process not only demonstrates the mechanical application of calculus rules but also enhances the understanding of how different functions interact when combined through operations like division. Furthermore, the simplification steps included in this guide are important for presenting the final answer in a concise and usable form, which is essential for further calculations or analyses that may be required in a broader context. Therefore, this detailed walkthrough aims to equip you with the skills and knowledge necessary to tackle similar differentiation problems with confidence and precision.
Understanding the Quotient Rule
Before we dive into differentiating F(x), it's imperative to grasp the quotient rule. The quotient rule is a vital tool in calculus, specifically designed for differentiating functions that are expressed as the ratio of two other functions. This rule is indispensable when dealing with expressions where one function is divided by another, a common scenario in various mathematical and scientific contexts. The quotient rule states that if we have a function F(x) defined as F(x) = u(x) / v(x), where u(x) and v(x) are differentiable functions, then the derivative of F(x), denoted as F'(x), can be calculated using the formula: F'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. This formula might seem complex at first glance, but it systematically guides us through the process of differentiating a quotient. The numerator of the formula involves the derivative of the numerator function (u'(x)) multiplied by the denominator function (v(x)), minus the original numerator function (u(x)) multiplied by the derivative of the denominator function (v'(x)). All of this is then divided by the square of the denominator function ([v(x)]^2). The quotient rule is not just a mathematical formula; it's a structured approach to handling division in calculus. By understanding and applying this rule correctly, we can accurately find the derivatives of complex rational functions, which are prevalent in fields like physics, engineering, and economics. Mastering the quotient rule is therefore a foundational skill for anyone working with calculus, enabling them to tackle a wider range of problems and apply mathematical principles effectively. In the subsequent sections, we will see how this rule is crucial in differentiating our given function, F(x) = e^(3x) / x^6.
Applying the Quotient Rule to F(x) = e^(3x) / x^6
Now, let's apply the quotient rule to our function, F(x) = e^(3x) / x^6. Identifying the components for the quotient rule is the initial step. We need to identify the numerator function, u(x), and the denominator function, v(x). In this case, u(x) = e^(3x) and v(x) = x^6. Next, we need to find the derivatives of both u(x) and v(x). To find u'(x), the derivative of e^(3x), we need to employ the chain rule. The chain rule is a fundamental concept in calculus used to differentiate composite functions, that is, functions that are nested within each other. In simpler terms, it's used when differentiating a function of a function. The chain rule states that if we have a composite function f(g(x)), its derivative is given by f'(g(x)) * g'(x). Applying the chain rule here, we differentiate the outer function, which is the exponential function, and then multiply by the derivative of the inner function, which is 3x. The derivative of e^(3x) is 3e^(3x). So, u'(x) = 3e^(3x). For v(x) = x^6, we use the power rule, which states that the derivative of x^n is nx^(n-1)*. Applying this, the derivative of x^6 is 6x^5. Thus, v'(x) = 6x^5. Now that we have u(x), v(x), u'(x), and v'(x), we can plug these into the quotient rule formula: F'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. Substituting the values, we get F'(x) = [x^6 * 3e^(3x) - e^(3x) * 6x^5] / (x6)2. This expression is the derivative of F(x), but it is not yet in its simplest form. The next step involves algebraic simplification to make the derivative more manageable and easier to interpret.
Simplifying the Derivative
After applying the quotient rule, our derivative looks like this: F'(x) = [x^6 * 3e^(3x) - e^(3x) * 6x^5] / (x6)2. The next crucial step is simplification. Simplifying the derivative not only makes it easier to work with but also presents the result in a cleaner, more understandable form. The first step in simplifying this expression is to look for common factors in the numerator. We can see that both terms in the numerator have e^(3x) and x^5 as common factors. Factoring out these common terms, we get: F'(x) = e^(3x) * x^5 * [3x - 6] / (x6)2. Now, let's simplify the denominator. We have (x6)2, which simplifies to x^12. So, our expression becomes: F'(x) = e^(3x) * x^5 * [3x - 6] / x^12. Next, we can simplify by canceling out x^5 from the numerator and denominator. This leaves us with: F'(x) = e^(3x) * [3x - 6] / x^7. We can further simplify the numerator by factoring out the constant 3: F'(x) = 3 * e^(3x) * [x - 2] / x^7. This is the simplified form of the derivative. It is much cleaner and easier to interpret than the initial expression we obtained after applying the quotient rule. Simplification is an essential skill in calculus as it allows us to handle derivatives more efficiently in subsequent calculations, such as finding critical points, inflection points, or analyzing the behavior of the function. Moreover, a simplified derivative is often necessary for further mathematical operations or for practical applications in various fields.
Final Result and Conclusion
In conclusion, after a detailed step-by-step process, we have successfully differentiated the function F(x) = e^(3x) / x^6. We began by identifying the need for the quotient rule, a fundamental concept in calculus for differentiating functions that are expressed as a ratio. We then identified the numerator and denominator of our function, u(x) = e^(3x) and v(x) = x^6, respectively. To find the derivative of u(x), we employed the chain rule, which is crucial for differentiating composite functions. This led us to find u'(x) = 3e^(3x). The derivative of v(x) was found using the power rule, resulting in v'(x) = 6x^5. Substituting these derivatives into the quotient rule formula, we obtained the initial derivative expression. The next critical step was simplification. We factored out common terms from the numerator and simplified the denominator, which involved using algebraic manipulation to reduce the complexity of the expression. This process led us to a more manageable form of the derivative. Finally, after factoring out the constant 3, we arrived at the simplified derivative: F'(x) = 3e^(3x) * (x - 2) / x^7. This final result represents the rate of change of the function F(x) with respect to x. It is now in a form that is both easy to understand and useful for further analysis, such as finding critical points, determining intervals of increase and decrease, and other applications in calculus. This detailed walkthrough demonstrates not only the application of differentiation rules but also the importance of simplification in calculus. The ability to simplify derivatives is crucial for practical applications and further mathematical operations. By mastering these techniques, one can effectively tackle a wide range of differentiation problems and apply them in various fields of study and real-world scenarios.
In summary, the derivative of F(x) = e^(3x) / x^6 is F'(x) = 3e^(3x) * (x - 2) / x^7.
