Derivative Of Y = 3ln(4lnx) A Step-by-Step Guide

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Introduction

In the realm of calculus, finding derivatives is a fundamental operation that allows us to understand the rate at which a function changes. Derivatives have wide-ranging applications in various fields, including physics, engineering, economics, and computer science. In this article, we will explore the process of finding the derivative of a specific function: y = 3ln(4lnx). This function involves nested logarithmic functions, making it a great example to illustrate the chain rule and other essential differentiation techniques. Understanding how to handle such functions is crucial for mastering calculus and its applications. So, let's embark on this journey of differentiation and unlock the secrets hidden within this intriguing function.

Understanding the Function

Before we dive into the differentiation process, it's crucial to understand the function we're dealing with: y = 3ln(4lnx). This function is a composition of several simpler functions, which means we'll need to apply the chain rule to find its derivative. Let's break it down:

  1. Innermost function: u = lnx. This is the natural logarithm of x, which is the logarithm to the base e (e ≈ 2.71828).
  2. Middle function: v = 4lnx = 4u. This is a constant multiple of the innermost function.
  3. Outermost function: y = 3ln(4lnx) = 3lnv. This is another natural logarithm, but this time it's applied to the middle function multiplied by a constant.

By recognizing these layers, we can see how the chain rule will come into play. The chain rule essentially states that the derivative of a composite function is the product of the derivatives of its individual components. We'll be working our way from the outside in, differentiating each layer with respect to its immediate input.

Applying the Chain Rule

The chain rule is a powerful tool for differentiating composite functions, and it's the key to finding the derivative of y = 3ln(4lnx). The chain rule can be expressed as:

dy/dx = (dy/du) * (du/dx)

where u is an intermediate function. In our case, we have multiple layers, so we'll need to extend the chain rule to handle them. Let's define our intermediate functions as we discussed earlier:

  • u = lnx
  • v = 4lnx = 4u
  • y = 3ln(4lnx) = 3lnv

Now we can apply the chain rule in a multi-layered fashion:

dy/dx = (dy/dv) * (dv/du) * (du/dx)

This formula tells us that we need to find the derivatives of y with respect to v, v with respect to u, and u with respect to x, and then multiply them together. Let's find each of these derivatives:

  1. du/dx: The derivative of u = lnx with respect to x is 1/x.
  2. dv/du: The derivative of v = 4u with respect to u is 4. This is because the derivative of a constant times a variable is just the constant.
  3. dy/dv: The derivative of y = 3lnv with respect to v is 3/ v. This follows from the rule that the derivative of lnv is 1/v, and we multiply by the constant 3.

Now we have all the pieces we need to apply the chain rule.

Calculating the Derivatives Step-by-Step

Let's calculate each derivative individually to make the process clear. This step-by-step approach will help solidify your understanding of the chain rule and how it applies to nested functions.

1. Derivative of u = ln(x)

The derivative of the natural logarithm function, ln(x), is a fundamental result in calculus. It's defined as:

d/dx(lnx) = 1/x

This is a standard derivative that you should memorize. It forms the basis for differentiating many logarithmic functions.

2. Derivative of v = 4ln(x)

To find the derivative of v = 4ln(x), we can use the constant multiple rule. This rule 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 4 and the function is ln(x). So,

d/dx(4lnx) = 4 * d/dx(lnx) = 4 * (1/x) = 4/x

3. Derivative of y = 3ln(4ln(x))

Here's where the chain rule comes into full effect. We've already identified the inner functions and their derivatives. Now, we differentiate the outermost function, y = 3ln(4ln(x)), with respect to its argument, which is 4ln(x). Let w = 4ln(x). Then y = 3ln(w), and

dy/dw = 3 * d/dw(lnw) = 3 * (1/w) = 3/(4lnx)

Now we need to multiply this by the derivative of w with respect to x, which we found in the previous step:

dw/dx = 4/x

Therefore, by the chain rule:

dy/dx = (dy/dw) * (dw/dx) = (3/(4lnx)) * (4/x) = 3/(xlnx)

Combining the Results

Now that we've calculated each individual derivative, we can combine them using the chain rule formula we established earlier:

dy/dx = (dy/dv) * (dv/du) * (du/dx)

We found that:

  • dy/dv = 3/v = 3/(4lnx)
  • dv/du = 4
  • du/dx = 1/x

Multiplying these together, we get:

dy/dx = (3/(4lnx)) * 4 * (1/x) = (3 * 4) / (4 * x * lnx) = 12 / (4xlnx)

Simplifying the expression by dividing both the numerator and denominator by 4, we arrive at the final derivative:

dy/dx = 3 / (xlnx)

This is the derivative of the function y = 3ln(4lnx) with respect to x. It represents the instantaneous rate of change of y as x varies.

Final Answer: The Derivative of y = 3ln(4lnx)

After meticulously applying the chain rule and simplifying the expression, we have arrived at the final answer. The derivative of the function y = 3ln(4lnx) with respect to x is:

dy/dx = 3 / (xlnx)

This result encapsulates the rate of change of the function at any given point x. It's a testament to the power and elegance of calculus in unraveling the behavior of complex functions. The chain rule, in particular, proves to be an indispensable tool when dealing with composite functions, allowing us to break down the differentiation process into manageable steps. This example not only demonstrates the application of the chain rule but also reinforces the importance of understanding the fundamental derivatives of logarithmic functions. Mastering these concepts is crucial for anyone venturing into the realms of calculus and its diverse applications.

Applications of Derivatives

Derivatives are not just abstract mathematical concepts; they have a wide range of practical applications in various fields. Understanding the derivative of a function allows us to analyze its behavior, such as finding its critical points, determining intervals of increase and decrease, and identifying concavity. Here are a few examples of how derivatives are used:

  1. Optimization: In optimization problems, we often need to find the maximum or minimum value of a function. Derivatives help us locate these points by finding where the function's slope is zero or undefined. This is crucial in fields like engineering, economics, and computer science, where optimizing resources or processes is essential.

  2. Physics: Derivatives are fundamental in physics for describing motion. Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. These concepts are used to model the motion of objects, from projectiles to planets.

  3. Economics: In economics, derivatives are used to analyze marginal cost, marginal revenue, and marginal profit. These concepts help businesses make decisions about production levels and pricing strategies. For example, a company might use derivatives to determine the production level that maximizes profit.

  4. Computer Science: Derivatives are used in machine learning algorithms, such as gradient descent, to optimize model parameters. Gradient descent is an iterative optimization algorithm that adjusts the parameters of a model to minimize a cost function. The derivative of the cost function is used to determine the direction in which to adjust the parameters.

  5. Curve Sketching: Derivatives provide valuable information about the shape of a function's graph. The first derivative tells us where the function is increasing or decreasing, and the second derivative tells us about its concavity. This information is used to sketch accurate graphs of functions.

  6. Related Rates: Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity. Derivatives are used to set up and solve these problems, which arise in various applications, such as calculating the rate at which a liquid is filling a container.

Conclusion

In this article, we successfully navigated the intricacies of differentiating the function y = 3ln(4lnx). We meticulously applied the chain rule, breaking down the function into its composite parts and differentiating each layer step by step. This process not only yielded the derivative, dy/dx = 3/(xlnx), but also provided a valuable illustration of the chain rule's power and versatility. Understanding and mastering the chain rule is paramount for tackling more complex differentiation problems in calculus. Furthermore, we highlighted the broad applications of derivatives across various disciplines, underscoring their significance in solving real-world problems.

Calculus, with its tools like derivatives, forms the bedrock of many scientific and engineering endeavors. By grasping these fundamental concepts and techniques, you equip yourself with the ability to analyze and model dynamic systems, optimize processes, and gain deeper insights into the world around us. The journey through calculus may present challenges, but the rewards of understanding and applying its principles are immeasurable. So, continue to explore, practice, and delve deeper into the fascinating realm of calculus, and you'll unlock a world of possibilities.