Derivative Of F(x) = 4 Sec(4x) A Step-by-Step Guide
Introduction
In the realm of calculus, finding the derivative of a function is a fundamental operation. The derivative, denoted as f'(x), represents the instantaneous rate of change of a function f(x) with respect to its input variable x. It provides valuable insights into the behavior of the function, such as its increasing or decreasing nature, concavity, and critical points. In this article, we delve into the process of finding the derivative of the function f(x) = 4 sec(4x). This function involves the secant trigonometric function, which requires specific differentiation rules and techniques. Understanding how to differentiate such functions is crucial for various applications in physics, engineering, and other scientific fields.
The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function, i.e., sec(x) = 1/cos(x). When differentiating secant functions, we often employ the chain rule, a powerful tool in calculus for differentiating composite functions. The chain rule states that the derivative of a composite function f(g(x)) is given by f'(g(x)) * g'(x). In our case, the function f(x) = 4 sec(4x) can be seen as a composite function, where the outer function is 4 sec(u) and the inner function is u = 4x. Applying the chain rule will be a key step in determining the derivative of f(x). Furthermore, knowing the derivative of the standard secant function, which is d/dx [sec(x)] = sec(x)tan(x), is essential for this calculation. With these principles in mind, we will systematically compute the derivative of f(x) = 4 sec(4x), providing a detailed explanation of each step involved. This article aims to enhance your understanding of differentiation techniques and their application to trigonometric functions.
Step-by-Step Differentiation of
To find the derivative f'(x) of the function f(x) = 4 sec(4x), we will employ the chain rule. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. A composite function is a function that is formed by combining two or more functions. In our case, f(x) = 4 sec(4x) can be viewed as a composite function where the outer function is 4 sec(u) and the inner function is u = 4x. The chain rule states that if we have a composite function f(g(x)), its derivative f'(x) is given by f'(g(x)) * g'(x). This means we need to differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to x. Let's break down the differentiation process step by step.
Step 1: Identify the Outer and Inner Functions
As mentioned earlier, we can identify the outer function as 4 sec(u) and the inner function as u = 4x. Here, u is an intermediate variable that simplifies the differentiation process. By recognizing these components, we can apply the chain rule more effectively. The outer function involves the secant function, which has a known derivative, and the inner function is a simple linear function, making its derivative straightforward to calculate. This decomposition is a critical first step in applying the chain rule, allowing us to address each part of the composite function separately and then combine the results correctly. Identifying the correct outer and inner functions is essential for the successful application of the chain rule.
Step 2: Differentiate the Outer Function with Respect to the Inner Function
The derivative of the outer function, 4 sec(u), with respect to u is given by 4 * d/du [sec(u)]. We know that the derivative of sec(u) with respect to u is sec(u)tan(u). Therefore, the derivative of the outer function is 4 sec(u)tan(u). This step involves applying the derivative rule for the secant function, which is a standard result in calculus. The constant multiple rule allows us to keep the constant 4 as a factor while differentiating sec(u). The resulting expression, 4 sec(u)tan(u), represents the rate of change of the outer function with respect to its input u. This is a key component in the chain rule, as it accounts for the transformation applied by the outer function.
Step 3: Differentiate the Inner Function with Respect to
The inner function is u = 4x. The derivative of u with respect to x, denoted as du/dx, is simply 4. This is a straightforward application of the power rule for differentiation, where the derivative of ax with respect to x is a. In this case, a is 4, so the derivative is 4. The derivative of the inner function represents the rate of change of u with respect to x. This component is crucial in the chain rule as it accounts for the transformation applied by the inner function. The simplicity of this derivative makes it easy to incorporate into the final result.
Step 4: Apply the Chain Rule
Now, we apply the chain rule, which states that f'(x) = f'(u) * du/dx. We have already found that f'(u) = 4 sec(u)tan(u) and du/dx = 4. Multiplying these together, we get f'(x) = 4 sec(u)tan(u) * 4. This step combines the results from the previous steps, bringing together the derivatives of the outer and inner functions. The chain rule effectively links these derivatives to give the overall derivative of the composite function. The resulting expression is a product of the derivatives, which accounts for how each part of the composite function contributes to the overall rate of change.
Step 5: Substitute Back
Finally, we substitute u = 4x back into the expression. This gives us f'(x) = 16 sec(4x)tan(4x). This substitution is necessary to express the derivative in terms of the original variable x. By replacing u with 4x, we obtain the final derivative of the function f(x). The resulting expression, 16 sec(4x)tan(4x), represents the instantaneous rate of change of f(x) = 4 sec(4x) with respect to x. This is the desired derivative, and it provides valuable information about the behavior of the function.
Therefore, the derivative of f(x) = 4 sec(4x) is:
f'(x) = 16 sec(4x)tan(4x)
Conclusion
In this article, we successfully found the derivative of the function f(x) = 4 sec(4x) using the chain rule. The chain rule is a powerful tool in calculus for differentiating composite functions, and its application here highlights its importance. By breaking down the function into its outer and inner components, we were able to systematically differentiate each part and combine the results to obtain the final derivative. The derivative, f'(x) = 16 sec(4x)tan(4x), provides valuable insights into the behavior of the function f(x). It represents the instantaneous rate of change of f(x) with respect to x, which is crucial for various applications in mathematics, physics, and engineering.
Understanding the process of differentiation, especially for trigonometric functions like secant, is essential for anyone studying calculus or related fields. The steps outlined in this article provide a clear and concise approach to finding derivatives of composite functions. The chain rule, combined with the knowledge of standard derivative rules, allows us to tackle complex differentiation problems effectively. The ability to differentiate functions like f(x) = 4 sec(4x) is a fundamental skill that opens doors to more advanced topics in calculus and its applications. This article aimed to enhance your understanding of differentiation techniques and their application to trigonometric functions, thereby strengthening your calculus foundation.
Furthermore, the derivative f'(x) = 16 sec(4x)tan(4x) can be used to analyze various properties of the function f(x) = 4 sec(4x), such as its increasing and decreasing intervals, critical points, and concavity. By setting the derivative equal to zero and solving for x, we can find the critical points of the function, which are potential local maxima or minima. The sign of the derivative in different intervals indicates whether the function is increasing or decreasing. Similarly, the second derivative can be used to determine the concavity of the function. Thus, the derivative is not just an abstract mathematical concept but a practical tool for analyzing the behavior of functions and solving real-world problems.
In summary, we have demonstrated how to find the derivative of f(x) = 4 sec(4x) using the chain rule, highlighting the importance of understanding and applying fundamental calculus concepts. The derivative f'(x) = 16 sec(4x)tan(4x) provides valuable information about the rate of change and behavior of the original function, making it a crucial tool in various mathematical and scientific applications. This step-by-step guide should serve as a useful resource for anyone looking to enhance their understanding of differentiation techniques and their applications.
