Calculate F'(2) For F(x) = E^(-1/x) A Step-by-Step Solution
In calculus, finding the derivative of a function is a fundamental operation. The derivative, denoted as f'(x), represents the instantaneous rate of change of the function f(x) with respect to its variable x. This article delves into the process of calculating the derivative of the function f(x) = e^(-1/x) and specifically evaluates it at x = 2. This problem showcases the application of the chain rule, a crucial concept in differential calculus. Understanding derivatives is essential for various applications in mathematics, physics, engineering, and economics, as it allows us to analyze the behavior of functions, such as finding their maximum and minimum values, determining their rates of change, and modeling real-world phenomena.
Differentiating exponential functions requires a solid understanding of the chain rule and the properties of exponents. The given function, f(x) = e^(-1/x), is a composite function, meaning it is a function within a function. The outer function is the exponential function e^u, and the inner function is u = -1/x. To find the derivative of such a function, we must apply the chain rule, which states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. This process might seem complex initially, but with a step-by-step approach and a clear understanding of the rules, it becomes manageable. This article aims to provide a detailed explanation of each step involved in finding the derivative and evaluating it at the specified point.
The calculation involves several key steps, including identifying the inner and outer functions, applying the chain rule, differentiating the inner and outer functions, and finally, substituting the value of x to find the derivative at that point. The chain rule is a cornerstone of calculus and is indispensable when dealing with composite functions. Failing to apply it correctly can lead to significant errors in the derivative. Moreover, a firm grasp of algebraic manipulations is crucial to simplify the expression and arrive at the correct answer. This article will provide a comprehensive guide through each step, ensuring clarity and understanding of the underlying concepts. Understanding this process is not only beneficial for solving this particular problem but also for tackling a wide range of derivative-related problems in calculus and beyond.
1. Understanding the Function
The given function is f(x) = e^(-1/x). We can rewrite this function as f(x) = e^u, where u = -1/x. This representation helps us identify the outer function (e^u) and the inner function (-1/x). Identifying the inner and outer functions is a crucial first step in applying the chain rule. Recognizing these components correctly ensures that we differentiate each part appropriately and combine them effectively. The outer function determines the overall structure of the derivative, while the inner function influences the rate of change within that structure. Therefore, a clear identification of these functions is paramount to a successful application of the chain rule.
2. Applying the Chain Rule
The chain rule states that if we have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x). In our case, f(u) = e^u and u(x) = -1/x. So, we need to find the derivatives of both e^u and -1/x. The chain rule is a powerful tool in calculus, allowing us to differentiate composite functions, which are functions formed by nesting one function inside another. Without the chain rule, differentiating such functions would be significantly more challenging, if not impossible. The chain rule effectively breaks down a complex differentiation problem into simpler, manageable parts. It is essential to understand the chain rule thoroughly to tackle a wide range of calculus problems, especially those involving exponential, logarithmic, and trigonometric functions.
3. Differentiating the Outer Function
The derivative of e^u with respect to u is simply e^u. This is a fundamental property of the exponential function. The exponential function's derivative being itself is a unique and important characteristic that simplifies many calculus problems. This property arises from the fundamental definition of the exponential function and its relationship to the natural logarithm. It is a cornerstone of calculus and is frequently used in various applications, including differential equations, growth and decay models, and probability theory. Understanding this property is essential for mastering calculus and its applications.
4. Differentiating the Inner Function
The derivative of u = -1/x with respect to x can be found by rewriting -1/x as -x^(-1). Using the power rule, the derivative is -(-1)x^(-2) = x^(-2) = 1/x². The power rule, which states that the derivative of x^n is nx^(n-1), is another fundamental tool in calculus. It allows us to differentiate polynomial functions and functions involving powers of x. In this case, applying the power rule to -1/x, which can be rewritten as -x^(-1), gives us the derivative 1/x^2. Mastering the power rule is crucial for differentiating a wide range of algebraic functions and is a foundational concept for more advanced calculus topics. Its application in this step demonstrates its versatility and importance in differential calculus.
5. Combining the Derivatives
Using the chain rule, f'(x) = (derivative of e^u) * (derivative of -1/x) = e^u * (1/x²). Now, substitute u = -1/x back into the equation: f'(x) = e^(-1/x) * (1/x²). Combining the derivatives obtained from the outer and inner functions using the chain rule gives us the complete derivative of the composite function. This step is where the individual components of the differentiation process come together to form the final expression for the derivative. The accuracy of this step is paramount, as any errors made earlier will propagate through the calculation and lead to an incorrect result. Therefore, careful attention to detail and a clear understanding of the chain rule are essential for this crucial step.
6. Evaluating at x = 2
Substitute x = 2 into f'(x): f'(2) = e^(-1/2) * (1/2²) = e^(-1/2) * (1/4). Since e^(-1/2) can be written as 1/√e, we have f'(2) = (1/4) * (1/√e). Multiplying the numerator and denominator by √e, we get f'(2) = √e / (4e). This can also be written as (1/4)√e, since e^(-1/2) is equivalent to 1/√e, which is the same as √e / e. Evaluating the derivative at a specific point, such as x = 2, gives us the instantaneous rate of change of the function at that particular point. This value is crucial for various applications, such as finding the slope of the tangent line to the function's graph at that point, determining the function's increasing or decreasing behavior, and optimizing functions in real-world scenarios. The evaluation step provides a concrete numerical value that can be interpreted and used for further analysis.
Therefore, f'(2) = (1/4)√e.
In conclusion, the derivative of the function f(x) = e^(-1/x) evaluated at x = 2 is f'(2) = (1/4)√e. This solution was obtained by applying the chain rule, differentiating the outer and inner functions, and then substituting the value of x. Understanding and applying the chain rule is crucial for solving problems involving composite functions in calculus. Mastering the chain rule is essential for success in calculus and related fields. It allows us to differentiate complex functions systematically and accurately. The ability to apply the chain rule effectively opens doors to solving a wide range of problems, from finding rates of change to optimizing functions. This skill is invaluable for anyone pursuing further studies in mathematics, physics, engineering, or any field that relies on mathematical modeling and analysis.
This problem serves as an excellent example of how to apply the chain rule in conjunction with the power rule and the properties of exponential functions. The detailed step-by-step solution provided in this article aims to enhance understanding and problem-solving skills in differential calculus. By breaking down the problem into manageable steps and providing clear explanations, this article serves as a valuable resource for students and anyone interested in deepening their understanding of calculus. The detailed solution not only provides the answer but also illustrates the thought process and techniques involved in solving such problems. This approach is crucial for developing a deeper understanding of the concepts and for building confidence in tackling similar problems in the future. By understanding the underlying principles and practicing the techniques, one can become proficient in calculus and its applications.
