Find The Derivative Dy/dx For The Functions Y = 2u³, U = 8x - 1
In calculus, the chain rule is a fundamental concept used to differentiate composite functions. A composite function is a function that is formed by applying one function to the result of another function. In simpler terms, it's a function within a function. The chain rule provides a method for finding the derivative of such composite functions. This article delves into a specific problem that exemplifies the application of the chain rule, offering a step-by-step solution and a detailed explanation to enhance understanding. Specifically, we will explore how to find dy/dx for the composite function y = 2u³ where u = 8x - 1. This problem is a classic example of how the chain rule works and is often encountered in introductory calculus courses. Mastering the chain rule is crucial for tackling more advanced calculus problems, especially those involving complex functions and implicit differentiation. So, let’s dive in and break down this problem to gain a solid understanding of this essential calculus tool.
Understanding the Chain Rule
Before we dive into the specifics of the problem, it's crucial to understand the chain rule itself. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically, if we have y = f(u) and u = g(x), then the derivative of y with respect to x (dy/dx) is given by:
dy/dx = (dy/du) * (du/dx)
This formula might seem daunting at first, but it's quite intuitive once you understand the underlying concept. Essentially, it says that to find the rate of change of y with respect to x, we need to consider how y changes with respect to u and how u changes with respect to x. The chain rule allows us to break down the differentiation process into smaller, more manageable steps. By understanding this fundamental principle, we can approach more complex problems with confidence. The chain rule is not just a formula to memorize; it's a powerful tool for understanding how different parts of a function interact and contribute to the overall rate of change. In the context of our problem, y = 2u³ is the outer function, and u = 8x - 1 is the inner function. We will use the chain rule to find the derivative of y with respect to x, step by step, to illustrate how this principle works in practice.
Problem Statement: Finding dy/dx
Now, let's restate the problem we aim to solve. We are given two functions:
- y = 2u³
- u = 8x - 1
Our task is to find dy/dx, which represents the derivative of y with respect to x. This means we want to determine how y changes as x changes. Since y is a function of u, and u is a function of x, we need to use the chain rule to connect these relationships. This problem is a perfect example of how the chain rule can be applied to find derivatives of composite functions. The chain rule allows us to break down a complex differentiation problem into smaller, more manageable steps. By finding the derivative of y with respect to u and the derivative of u with respect to x, we can then multiply these derivatives together to find dy/dx. This step-by-step approach not only makes the problem easier to solve but also provides a deeper understanding of the chain rule itself. The problem is not just about finding the correct answer; it's about understanding the process and the logic behind it.
Step-by-Step Solution
To find dy/dx, we will follow these steps:
Step 1: Find dy/du
First, we need to differentiate y with respect to u. Given y = 2u³, we apply the power rule, which states that the derivative of xⁿ with respect to x is nxⁿ⁻¹. Applying this rule to our function, we get:
dy/du = d/du (2u³) = 2 * 3u² = 6u²
This step involves a straightforward application of the power rule, a fundamental concept in calculus. The derivative dy/du represents the rate of change of y with respect to u. In other words, it tells us how much y changes for a small change in u. This is an essential component of the chain rule, as it links the change in the outer function (y) to the change in its intermediate variable (u). Understanding this step is crucial for grasping the chain rule's mechanics. We are essentially finding the rate at which y changes as u varies, which is a necessary piece of information for finding the overall rate of change of y with respect to x. This step highlights the beauty of calculus, where complex problems can be broken down into smaller, more manageable parts, each of which can be solved using basic differentiation rules.
Step 2: Find du/dx
Next, we need to differentiate u with respect to x. Given u = 8x - 1, we differentiate with respect to x:
du/dx = d/dx (8x - 1) = 8
Here, we find the derivative of u with respect to x, which represents the rate of change of u as x changes. The derivative of 8x is 8, and the derivative of the constant -1 is 0. Therefore, du/dx is simply 8. This means that for every unit change in x, u changes by 8 units. This is another crucial component of the chain rule, as it links the change in the intermediate variable (u) to the change in the independent variable (x). Understanding this relationship is vital for applying the chain rule correctly. We are essentially finding the rate at which u changes as x varies, which is the final piece of information we need to calculate dy/dx. This step demonstrates the power of calculus in describing relationships between variables and how they change with respect to each other.
Step 3: Apply the Chain Rule
Now, we apply the chain rule formula: dy/dx = (dy/du) * (du/dx). We have already found dy/du = 6u² and du/dx = 8. Multiplying these together, we get:
dy/dx = (6u²) * (8) = 48u²
This step is the heart of the chain rule application. We are combining the two derivatives we found in the previous steps to find the overall rate of change of y with respect to x. By multiplying dy/du and du/dx, we are essentially linking the rate of change of y with respect to u to the rate of change of u with respect to x. This is the essence of the chain rule: it allows us to find the derivative of a composite function by considering the derivatives of its individual components. The result, dy/dx = 48u², represents the rate of change of y with respect to x in terms of u. However, we want to express dy/dx in terms of x, so we need to substitute u in terms of x, which is the next step in our solution.
Step 4: Substitute u
Finally, we substitute u = 8x - 1 back into the equation for dy/dx:
dy/dx = 48(8x - 1)²
This final step completes the solution. We have successfully found dy/dx in terms of x. By substituting u with its expression in terms of x, we have expressed the derivative of y with respect to x solely in terms of x. This is often the desired form for the derivative, as it allows us to directly calculate the rate of change of y for any given value of x. The final answer, dy/dx = 48(8x - 1)², represents the instantaneous rate of change of y with respect to x. This means that for a small change in x, the change in y is approximately 48(8x - 1)² times the change in x. This step highlights the power of algebraic manipulation in simplifying and expressing mathematical results in their most useful form.
The Answer
Therefore, the derivative of y with respect to x is:
dy/dx = 48(8x - 1)²
So, the correct answer is (a) 48(8x-1)². This solution demonstrates the step-by-step application of the chain rule, a fundamental concept in calculus. By breaking down the problem into smaller, more manageable steps, we were able to find the derivative of a composite function. This approach not only provides the correct answer but also reinforces the understanding of the underlying principles. The chain rule is a powerful tool for differentiating complex functions, and mastering its application is essential for success in calculus. This problem serves as a valuable example of how the chain rule works in practice, highlighting the importance of understanding each step and the logic behind it. The ability to apply the chain rule correctly is a crucial skill for tackling more advanced calculus problems.
Conclusion
In conclusion, we have successfully found dy/dx for the given functions y = 2u³ and u = 8x - 1 using the chain rule. The step-by-step solution involved finding dy/du, du/dx, applying the chain rule formula, and substituting u back in terms of x. This process highlights the power and elegance of the chain rule in differentiating composite functions. The chain rule is not just a formula; it's a fundamental concept that allows us to understand how different parts of a function interact and contribute to its overall rate of change. By mastering the chain rule, we can tackle more complex differentiation problems with confidence. This problem serves as a valuable learning experience, reinforcing the importance of breaking down complex problems into smaller, more manageable steps. The ability to apply the chain rule correctly is a crucial skill for anyone studying calculus and related fields. This article has provided a comprehensive guide to solving this particular problem, but the principles and techniques discussed can be applied to a wide range of differentiation problems. Remember, practice is key to mastering calculus concepts, so continue to work through examples and challenge yourself with increasingly complex problems. By doing so, you will develop a deeper understanding of calculus and its applications.