Finding The Derivative Of Y=(3x-1)³ A Step By Step Solution

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#Understanding the Core Concept of Derivatives

Before we dive into the specifics of finding the derivative of y = (3x - 1)³, it's crucial to grasp the fundamental concept of derivatives in calculus. In essence, a derivative measures the instantaneous rate of change of a function. Imagine a curve plotted on a graph; the derivative at a particular point on that curve represents the slope of the line tangent to the curve at that point. This slope tells us how the function's output changes in response to a tiny change in its input.

Derivatives are the cornerstone of differential calculus, playing a pivotal role in various fields, from physics and engineering to economics and computer science. They enable us to analyze motion, optimize processes, model complex systems, and solve a myriad of real-world problems. For instance, in physics, derivatives help us determine the velocity and acceleration of an object given its position function. In economics, they are used to calculate marginal cost and marginal revenue, aiding in decision-making processes. The beauty of derivatives lies in their ability to capture dynamic behavior and provide insights into how things change over time or with respect to other variables.

To truly understand derivatives, it's essential to familiarize yourself with the notation and terminology used. The derivative of a function y with respect to a variable x is commonly denoted as dy/dx or y'. This notation signifies the infinitesimal change in y divided by the infinitesimal change in x. The process of finding a derivative is called differentiation. There are various rules and techniques for differentiation, each applicable to different types of functions. These rules, such as the power rule, product rule, quotient rule, and chain rule, provide a systematic approach to finding derivatives of complex expressions. As we delve into the derivative of y = (3x - 1)³, we will be utilizing one of these crucial rules – the chain rule.

#Deciphering the Problem: y = (3x-1)³

At the heart of our exploration lies the function y = (3x - 1)³. This function represents a polynomial expression raised to a power. To effectively find its derivative, we need to recognize its composite nature. The expression (3x - 1) is nested within the cubing operation. This is where the chain rule, a fundamental concept in calculus, comes into play. The chain rule provides a methodical way to differentiate composite functions – functions that are composed of other functions.

Consider the structure of y = (3x - 1)³. We can think of it as an outer function, which is the cubing operation (something raised to the power of 3), and an inner function, which is the linear expression (3x - 1). The chain rule essentially states that to find the derivative of a composite function, 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 the independent variable (in this case, x). This might sound complex, but we will break it down step by step.

The ability to recognize composite functions is a crucial skill in calculus. It allows us to apply the appropriate differentiation techniques and avoid common pitfalls. Without the chain rule, differentiating expressions like y = (3x - 1)³ would be significantly more challenging. The chain rule provides a streamlined approach, ensuring we account for the interconnectedness of the inner and outer functions. Understanding the chain rule is not only essential for this specific problem but also for a vast array of differentiation problems in calculus and beyond. It's a tool that empowers us to tackle complex functions with confidence and accuracy.

The chain rule is a cornerstone of differential calculus, specifically designed to handle the differentiation of composite functions. A composite function is essentially a function within a function – where the output of one function serves as the input of another. Understanding and applying the chain rule correctly is paramount for solving a wide range of calculus problems.

In mathematical terms, the chain rule can be expressed as follows: If we have a function y = f(g(x)), where f is the outer function and g(x) is the inner function, then the derivative of y with respect to x, denoted as dy/dx, is given by:

dy/dx = f'(g(x)) * g'(x)

Let's break down this formula piece by piece:

  • f'(g(x)) represents the derivative of the outer function f evaluated at the inner function g(x). In simpler terms, we differentiate the outer function while keeping the inner function intact.
  • g'(x) represents the derivative of the inner function g(x) with respect to x. This is the standard derivative of the inner function.
  • The multiplication of these two derivatives gives us the overall derivative of the composite function.

The chain rule can be visualized as a step-by-step process: First, we differentiate the "outer layer" of the function, treating the "inner layer" as a single entity. Then, we multiply by the derivative of that "inner layer." This process ensures that we account for the contribution of each function to the overall rate of change. Mastering the chain rule unlocks our ability to differentiate complex expressions that would otherwise be intractable. It's a tool that empowers us to delve deeper into the world of calculus and tackle increasingly challenging problems.

#Step-by-Step Solution: Applying the Chain Rule to y = (3x-1)³

Now, let's put the chain rule into action and find the derivative of our function, y = (3x - 1)³. To begin, we need to identify the outer and inner functions. As discussed earlier, the outer function is the cubing operation, and the inner function is the linear expression (3x - 1). We can represent this more formally as:

  • Outer function: f(u) = u³
  • Inner function: g(x) = 3x - 1

Where u is a placeholder representing the inner function.

Now, we proceed with the differentiation process step by step:

  1. Differentiate the outer function: We need to find the derivative of f(u) = u³ with respect to u. Using the power rule of differentiation (which states that the derivative of xⁿ is nxⁿ⁻¹), we get:

    f'(u) = 3u²

  2. Differentiate the inner function: We need to find the derivative of g(x) = 3x - 1 with respect to x. The derivative of 3x is 3, and the derivative of the constant -1 is 0. Therefore:

    g'(x) = 3

  3. Apply the chain rule: Now, we combine the derivatives we found in the previous steps using the chain rule formula: dy/dx = f'(g(x)) * g'(x). We substitute g(x), which is (3x - 1), into f'(u), and then multiply by g'(x):

    dy/dx = 3(3x - 1)² * 3

  4. Simplify the expression: Finally, we simplify the expression by multiplying the constants:

    dy/dx = 9(3x - 1)²

Thus, the derivative of y = (3x - 1)³ is 9(3x - 1)². This step-by-step application of the chain rule demonstrates the systematic approach to differentiating composite functions. By breaking down the problem into manageable parts and applying the appropriate rules, we can confidently arrive at the correct solution. The ability to apply the chain rule effectively is a valuable asset in calculus and beyond.

#Selecting the Correct Answer

Having meticulously calculated the derivative of y = (3x - 1)³ as dy/dx = 9(3x - 1)², we can now confidently identify the correct answer from the given options. Let's revisit the options:

  • A. 3(3x - 1)²
  • B. (3x - 1)²
  • C. 3
  • D. 9(3x - 1)²

By comparing our calculated derivative with the provided options, it becomes clear that option D, 9(3x - 1)², is the correct answer. The other options represent incorrect applications of the differentiation rules or missed steps in the chain rule process. Option A is missing a factor of 3. Option B is missing the constant factor 9. Option C is simply incorrect and doesn't reflect the derivative of the given function.

This exercise highlights the importance of a thorough and systematic approach to differentiation. By carefully applying the chain rule and simplifying the resulting expression, we can arrive at the correct answer and avoid common pitfalls. The ability to accurately identify and select the correct answer is a testament to a solid understanding of calculus principles and problem-solving skills.

#Conclusion: Mastering Derivatives for Mathematical Success

In conclusion, the derivative of y = (3x - 1)³ is 9(3x - 1)², a result obtained by carefully applying the chain rule. This exercise serves as a valuable illustration of the power and importance of the chain rule in differential calculus. Mastering the chain rule, along with other differentiation techniques, is crucial for success in calculus and related fields.

Derivatives are not merely abstract mathematical concepts; they are powerful tools with widespread applications in various disciplines. From physics and engineering to economics and computer science, derivatives enable us to model dynamic systems, optimize processes, and solve real-world problems. A solid understanding of derivatives opens doors to advanced concepts in mathematics and provides a foundation for further exploration in scientific and technological fields.

The ability to differentiate functions accurately and efficiently is a fundamental skill for any aspiring mathematician, scientist, or engineer. The process involves not only knowing the rules of differentiation but also understanding when and how to apply them. Practice is key to mastering these skills. Working through a variety of examples, from simple to complex, helps solidify understanding and build confidence.

By tackling problems like finding the derivative of y = (3x - 1)³, we not only hone our differentiation skills but also gain a deeper appreciation for the elegance and power of calculus. The journey of learning calculus is a rewarding one, filled with intellectual challenges and exciting discoveries. With dedication and perseverance, anyone can unlock the beauty and utility of derivatives and other calculus concepts.