Differentiating Sin(8ln(x)) A Step-by-Step Tutorial With Chain Rule
This comprehensive tutorial provides a step-by-step guide on how to differentiate the function f(x) = sin(8 ln(x)). We will utilize the Chain Rule, a fundamental concept in calculus, to accurately find the derivative. This exercise is crucial for understanding how to differentiate composite functions, which are functions within functions. Mastering this skill is essential for tackling more complex calculus problems.
Step 1: Understanding the Chain Rule
Chain Rule is the key to differentiating composite functions. Recall that the Chain Rule, in terms of functions h(x) and g(x), states that if g is differentiable at x and h is differentiable at g(x), then the composite function h(g(x)) is differentiable at x, and its derivative is given by:
In simpler terms, the Chain Rule tells us to differentiate the outer function, keeping the inner function the same, and then multiply by the derivative of the inner function. This process is like peeling an onion, layer by layer. We first address the outermost layer and then delve into the subsequent layers until we reach the core. In the context of our problem, the outer function is the sine function (sin), and the inner function is 8ln(x). Understanding this concept is paramount to correctly applying the Chain Rule and arriving at the accurate derivative. The Chain Rule is not just a formula; it's a systematic approach to differentiating complex functions. It allows us to break down the problem into manageable parts, making the differentiation process much more straightforward. Ignoring the Chain Rule when dealing with composite functions will invariably lead to incorrect results. Therefore, a firm grasp of this rule is non-negotiable for any calculus student.
Step 2: Identifying the Inner and Outer Functions
Before applying the Chain Rule, we must clearly identify the inner and outer functions in our given function, f(x) = sin(8 ln(x)). This identification is crucial for the correct application of the Chain Rule. In this case, the outer function, which we'll call h(u), is the sine function: h(u) = sin(u). The inner function, which we'll call g(x), is 8 ln(x). Think of it as plugging the inner function, 8 ln(x), into the outer function, sin(u), where 'u' is replaced by 8 ln(x). Therefore, f(x) can be seen as h(g(x)), a composite function. The ability to correctly decompose a composite function into its inner and outer components is a foundational skill in calculus. Without this skill, the Chain Rule cannot be effectively applied. Misidentifying the inner and outer functions will inevitably lead to an incorrect derivative. This step is not just about recognizing the functions; it's about understanding the order of operations within the function. The inner function is what's being acted upon first, and the outer function is the operation being applied to the result of the inner function. This perspective is key to mastering the Chain Rule and differentiating a wide range of composite functions.
Step 3: Differentiating the Outer Function
Now that we've identified the outer function as h(u) = sin(u), we need to find its derivative, h'(u). The derivative of the sine function is a fundamental calculus concept. Recall that the derivative of sin(u) with respect to u is cos(u). Therefore, h'(u) = cos(u). This is a direct application of a basic differentiation rule. It's important to have these fundamental derivatives memorized, as they form the building blocks for differentiating more complex functions. In the context of the Chain Rule, we'll need to evaluate this derivative at the inner function, g(x), which is 8 ln(x). This means we'll substitute 'u' with '8 ln(x)' in h'(u). This step highlights the importance of understanding the Chain Rule's structure. We're not just finding the derivative of the outer function in isolation; we're finding it as it acts upon the inner function. This subtle but critical distinction is what makes the Chain Rule so powerful and versatile. A solid understanding of trigonometric derivatives is crucial for success in calculus, and this step provides a practical application of that knowledge.
Step 4: Differentiating the Inner Function
Next, we need to differentiate the inner function, g(x) = 8 ln(x). This involves applying the rules of differentiation to a logarithmic function. Recall that the derivative of ln(x) with respect to x is 1/x. Also, remember the constant multiple rule, which states that the derivative of a constant times a function is the constant times the derivative of the function. In this case, the constant is 8. Therefore, the derivative of 8 ln(x) with respect to x, denoted as g'(x), is 8 * (1/x) = 8/x. This step demonstrates the application of multiple differentiation rules in conjunction. It's not just about knowing the derivative of ln(x); it's about knowing how to apply it in the presence of a constant multiple. This ability to combine different differentiation rules is a hallmark of a proficient calculus student. Understanding the properties of logarithms is also essential for this step. The natural logarithm, ln(x), has a unique derivative that makes it a fundamental function in calculus. This step reinforces the importance of mastering the derivatives of basic functions as a foundation for tackling more complex problems.
Step 5: Applying the Chain Rule
Now we have all the pieces necessary to apply the Chain Rule. We found h'(u) = cos(u) and g'(x) = 8/x. The Chain Rule states that the derivative of f(x) = h(g(x)) is f'(x) = h'(g(x)) * g'(x). Substituting the functions we found, we get f'(x) = cos(8 ln(x)) * (8/x). This is the final step where we combine the derivatives of the inner and outer functions according to the Chain Rule's formula. It's crucial to ensure that the substitution is done correctly, replacing 'u' in h'(u) with the inner function g(x). This step encapsulates the essence of the Chain Rule – it's a multiplicative relationship between the derivative of the outer function evaluated at the inner function and the derivative of the inner function itself. This step also highlights the importance of careful notation and organization. Keeping track of the different functions and their derivatives is essential for avoiding errors. The final expression, cos(8 ln(x)) * (8/x), is the derivative of the original function, f(x) = sin(8 ln(x)).
Step 6: Simplifying the Result
While the previous step provided the correct derivative, it's often beneficial to simplify the expression for clarity and ease of use. Our derivative is f'(x) = cos(8 ln(x)) * (8/x). We can rewrite this as f'(x) = (8 * cos(8 ln(x))) / x. This simplification involves basic algebraic manipulation, specifically multiplying the constant 8 by the cosine term and placing the entire expression over the denominator x. This simplified form is more compact and easier to work with in subsequent calculations or applications. Simplification is a valuable skill in calculus, as it allows us to present results in a clear and concise manner. A simplified expression is often easier to interpret and use in further analysis or problem-solving. While not always strictly necessary, simplifying the derivative is generally considered good practice. It demonstrates a thorough understanding of the material and an attention to detail. In this case, the simplification is straightforward, but in more complex scenarios, it may involve more advanced algebraic techniques. The final simplified derivative, f'(x) = (8 * cos(8 ln(x))) / x, represents the rate of change of the original function, f(x) = sin(8 ln(x)), with respect to x.
Therefore, the derivative of the function f(x) = sin(8 ln(x)) is:
This tutorial has walked you through each step of differentiating a composite function using the Chain Rule. By understanding and practicing these steps, you can confidently tackle similar differentiation problems.
