Derivative Of F(x) = 5 Arctan(7 Sin(4x)) A Calculus Exploration

Introduction

In the realm of calculus, finding the derivatives of complex functions often requires a meticulous application of various differentiation rules. This article delves into the process of finding the derivative of a composite function, specifically $f(x) = 5 \arctan(7 \sin(4x))$. This problem showcases the power of the chain rule and the derivatives of trigonometric and inverse trigonometric functions. We will embark on a step-by-step journey, unraveling the layers of this function to reveal its derivative. Understanding the nuances of such composite functions is crucial for mastering calculus and its applications in various scientific and engineering fields. So, let's begin our exploration into the fascinating world of derivatives and composite functions!

The main goal of this exploration is to find $f'(x)$, the derivative of the given function. To achieve this, we will employ the chain rule, a fundamental concept in calculus that allows us to differentiate composite functions. The chain rule essentially states that the derivative of a composite function is the product of the derivatives of its outer and inner functions. In our case, we have a function composed of the arctangent function, the sine function, and a linear function. By systematically applying the chain rule, we will break down the function into its constituent parts, differentiate each part, and then combine the results to obtain the final derivative. This process will not only provide us with the answer but also enhance our understanding of how different differentiation rules work in tandem. As we proceed, we will emphasize the importance of careful application and attention to detail in calculus problems. This step-by-step approach will help clarify the concepts and solidify our grasp on the techniques involved.

Before we dive into the specific steps, let's briefly review the essential differentiation rules that will be our guiding stars in this endeavor. First and foremost, we need to recall the derivative of the arctangent function, which is a cornerstone of this problem. The derivative of $\arctan(u)$ with respect to $u$ is given by $\frac{1}{1 + u^2}$. This rule will be crucial when we tackle the outermost layer of our composite function. Next, we'll need the derivative of the sine function. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$. This trigonometric derivative will come into play when we differentiate the inner sine function. Finally, we'll use the power of the chain rule itself, which, as we mentioned earlier, is the linchpin of differentiating composite functions. The chain rule allows us to handle functions within functions, ensuring we account for the derivatives of each layer. With these tools at our disposal, we're well-equipped to tackle the challenge of finding $f'(x)$.

Applying the Chain Rule: A Step-by-Step Differentiation

Now, let's begin the process of finding the derivative, $f'(x)$, by systematically applying the chain rule. The given function is $f(x) = 5 \arctan(7 \sin(4x))$. The first step is to identify the outermost function, which in this case is the arctangent function. We know that the derivative of $5 \arctan(u)$ with respect to $u$ is $\frac{5}{1 + u^2}$. Here, $u$ represents the inner function, which is $7 \sin(4x)$. So, the first application of the chain rule gives us:

$f'(x) = 5 \cdot \frac{1}{1 + (7 \sin(4x))^2} \cdot \frac{d}{dx}(7 \sin(4x))$

This step highlights the essence of the chain rule: differentiating the outer function while keeping the inner function intact, and then multiplying by the derivative of the inner function. The constant factor of 5 is simply carried through the differentiation process, as it is a multiplicative constant. The next challenge is to find the derivative of the inner function, $7 \sin(4x)$. This involves another application of the chain rule, as we have yet another composite function.

Next, we need to differentiate $7 \sin(4x)$. This function is composed of the sine function and a linear function, $4x$. The derivative of $7 \sin(u)$ with respect to $u$ is $7 \cos(u)$, where $u$ in this case is $4x$. Thus, we have:

$\frac{d}{dx}(7 \sin(4x)) = 7 \cos(4x) \cdot \frac{d}{dx}(4x)$

Here, we have again applied the chain rule. We differentiated the sine function, obtaining the cosine function, and kept the inner function, $4x$, intact. We then multiplied by the derivative of the inner function, $\frac{d}{dx}(4x)$. This step-by-step approach is crucial to avoid errors and maintain clarity in the differentiation process. The derivative of $4x$ with respect to $x$ is simply 4. This is a straightforward application of the power rule of differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$. In this case, $n=1$, so the derivative of $4x$ is $4 \cdot 1 \cdot x^0 = 4$. Therefore, we can substitute this result back into our expression:

$\frac{d}{dx}(7 \sin(4x)) = 7 \cos(4x) \cdot 4 = 28 \cos(4x)$

Now we have successfully differentiated the inner function, $7 \sin(4x)$. This result is a critical piece of the puzzle, as we need it to complete the derivative of the original function. We can now substitute this back into our expression for $f'(x)$ that we derived in the first step. This substitution will give us a more complete picture of the derivative of the composite function. The process of breaking down the function into smaller, more manageable parts and then combining the results is a testament to the power and elegance of the chain rule.

Combining the Results: The Final Derivative

Now that we have found the derivative of the inner function, we can substitute it back into our initial expression for $f'(x)$. Recall that we had:

$f'(x) = 5 \cdot \frac{1}{1 + (7 \sin(4x))^2} \cdot \frac{d}{dx}(7 \sin(4x))$

And we found that:

$\frac{d}{dx}(7 \sin(4x)) = 28 \cos(4x)$

Substituting this result into the expression for $f'(x)$, we get:

$f'(x) = 5 \cdot \frac{1}{1 + (7 \sin(4x))^2} \cdot 28 \cos(4x)$

This is a significant step towards our final answer. We have now successfully differentiated all the components of the composite function and combined them using the chain rule. The next step is to simplify this expression to obtain the most concise form of the derivative. Simplification not only makes the expression easier to work with but also reveals the underlying structure of the derivative. This is an important aspect of calculus, as simplified expressions can often provide deeper insights into the behavior of the function. In our case, the simplification involves combining the constant terms and simplifying the denominator.

To simplify, we first multiply the constants in the numerator: $5 \cdot 28 = 140$. Then, we simplify the term $(7 \sin(4x))^2$ in the denominator. This gives us $49 \sin^2(4x)$. Substituting these simplifications into the expression for $f'(x)$, we have:

$f'(x) = \frac{140 \cos(4x)}{1 + 49 \sin^2(4x)}$

This is the final simplified form of the derivative. We have successfully navigated the complexities of the composite function and arrived at a clear and concise expression for its derivative. This result encapsulates the rate of change of the function $f(x)$ with respect to $x$. The derivative provides valuable information about the function's behavior, such as its increasing and decreasing intervals, critical points, and concavity. The process of finding this derivative has reinforced our understanding of the chain rule, the derivatives of trigonometric functions, and the importance of careful simplification. Now, we can confidently say that we have successfully tackled this challenging calculus problem.

Conclusion

In this comprehensive exploration, we have successfully determined the derivative of the composite function $f(x) = 5 \arctan(7 \sin(4x))$ to be $f'(x) = \frac{140 \cos(4x)}{1 + 49 \sin^2(4x)}$. This journey has provided us with a clear illustration of the power and elegance of the chain rule in calculus. The chain rule allowed us to systematically break down a complex function into its constituent parts, differentiate each part, and then combine the results to obtain the final derivative. This step-by-step approach is a cornerstone of calculus and is essential for tackling a wide range of differentiation problems. By mastering the chain rule, we have expanded our calculus toolkit and gained a deeper understanding of how functions interact with each other.

Throughout this process, we have emphasized the importance of understanding the underlying concepts and applying them methodically. Each step, from identifying the outermost function to simplifying the final expression, was carefully explained and justified. This meticulous approach is crucial for avoiding errors and ensuring accuracy in calculus problems. Furthermore, we highlighted the significance of simplifying the final result. A simplified expression not only makes the derivative easier to work with but also provides valuable insights into the function's behavior. The ability to simplify expressions is a hallmark of mathematical proficiency and is highly valued in various applications of calculus.

The derivative we obtained, $f'(x) = \frac{140 \cos(4x)}{1 + 49 \sin^2(4x)}$, provides a wealth of information about the original function $f(x)$. For instance, we can use this derivative to analyze the critical points of $f(x)$, which are the points where the derivative is either zero or undefined. These critical points can help us identify the local maxima and minima of the function. Additionally, we can use the derivative to determine the intervals where the function is increasing or decreasing. By examining the sign of the derivative, we can infer whether the function is rising or falling. This kind of analysis is fundamental in many applications of calculus, such as optimization problems and curve sketching. The derivative also plays a crucial role in understanding the concavity of the function, which describes how the function curves. By analyzing the second derivative, we can identify intervals where the function is concave up or concave down. This information, combined with the critical points and increasing/decreasing intervals, provides a comprehensive understanding of the function's shape and behavior.

In conclusion, finding the derivative of $f(x) = 5 \arctan(7 \sin(4x))$ has been a rewarding exercise in applying the chain rule and other fundamental calculus concepts. This process has not only provided us with a concrete result but has also deepened our understanding of differentiation techniques and their applications. The derivative, $f'(x) = \frac{140 \cos(4x)}{1 + 49 \sin^2(4x)}$, stands as a testament to the power of calculus in unraveling the complexities of composite functions. As we continue our journey in calculus, the lessons learned from this exploration will undoubtedly serve us well in tackling even more challenging problems.