Finding The Derivative Of Y = 3 / (2x)^-2 A Step-by-Step Guide

In the realm of calculus, finding the derivative of a function is a fundamental operation. The derivative, often denoted as y' or dy/dx, represents the instantaneous rate of change of a function with respect to its independent variable. It is a cornerstone concept in various fields, including physics, engineering, economics, and computer science. Mastering the techniques for finding derivatives is crucial for understanding and applying calculus in real-world scenarios. This article delves into the process of finding the derivative of a function, focusing on a specific example that involves rewriting, differentiating, and simplifying the function. We will guide you through each step of the process, providing a comprehensive understanding of the underlying principles and techniques.

Before we delve into the specific example, it is essential to grasp the fundamental concept of derivatives. The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point. It quantifies how much the function's output changes for a small change in its input. In other words, it measures the instantaneous rate of change. The derivative of a function f(x) is denoted as f'(x) or dy/dx, where y = f(x). The process of finding the derivative is called differentiation. There are various rules and techniques for differentiation, each applicable to different types of functions. These rules include the power rule, the constant multiple rule, the sum/difference rule, the product rule, the quotient rule, and the chain rule. Understanding these rules is crucial for effectively differentiating a wide range of functions.

Let's consider the function y = 3 / (2x)^-2. Our goal is to find the derivative of this function, y'. To achieve this, we will follow a systematic approach that involves rewriting the function, differentiating it using appropriate rules, and simplifying the result. This step-by-step process will not only lead us to the solution but also provide a clear understanding of the techniques involved.

Step 1: Rewrite the Function

The first step in finding the derivative is to rewrite the function in a more convenient form for differentiation. The given function, y = 3 / (2x)^-2, involves a negative exponent in the denominator. To simplify this, we can use the property that a^(-n) = 1 / a^n. Applying this property, we can rewrite (2x)^-2 as 1 / (2x)^2. Therefore, the function becomes:

y = 3 / (1 / (2x)^2)

To further simplify, we can multiply the numerator and denominator by (2x)^2, which gives us:

y = 3 * (2x)^2

Now, we can expand (2x)^2 as 4x^2, resulting in:

y = 3 * 4x^2

Finally, multiplying the constants, we get the rewritten function:

y = 12x^2

This rewritten form is much easier to differentiate compared to the original form. It allows us to directly apply the power rule, which is a fundamental rule for differentiating power functions.

Step 2: Differentiate the Rewritten Function

Now that we have rewritten the function as y = 12x^2, we can proceed with differentiation. The power rule states that if y = ax^n, where a is a constant and n is a real number, then the derivative y' is given by:

y' = n * ax^(n-1)

In our case, a = 12 and n = 2. Applying the power rule, we get:

y' = 2 * 12x^(2-1)

Simplifying this expression, we have:

y' = 24x^1

Which is simply:

y' = 24x

Therefore, the derivative of the rewritten function y = 12x^2 is y' = 24x. This step demonstrates the power of rewriting functions before differentiation, as it simplifies the process and allows us to directly apply differentiation rules.

Step 3: Simplify the Derivative (If Necessary)

In this particular case, the derivative y' = 24x is already in a simplified form. There are no further algebraic manipulations needed to make it more concise or understandable. However, in some cases, the derivative obtained after differentiation may need further simplification. This could involve combining like terms, factoring out common factors, or using trigonometric identities. The goal of simplification is to express the derivative in its most compact and manageable form. In our example, the derivative y' = 24x is already in its simplest form, so we can conclude that the derivative of the original function y = 3 / (2x)^-2 is y' = 24x.

In this article, we have demonstrated the process of finding the derivative of a function by rewriting, differentiating, and simplifying. We considered the example function y = 3 / (2x)^-2 and systematically worked through each step. First, we rewrote the function as y = 12x^2 by applying the properties of exponents. Then, we differentiated the rewritten function using the power rule, obtaining y' = 24x. Finally, we observed that the derivative was already in its simplest form. This example highlights the importance of rewriting functions before differentiation to simplify the process. It also reinforces the application of the power rule, a fundamental rule in calculus. By mastering these techniques, you can effectively find the derivatives of a wide range of functions.

Finding the derivative of a function is a crucial skill in calculus with wide-ranging applications. This article has provided a step-by-step guide to finding the derivative of a function, focusing on the example of y = 3 / (2x)^-2. We emphasized the importance of rewriting the function in a convenient form before differentiation, applying appropriate differentiation rules, and simplifying the result. The process of rewriting, differentiating, and simplifying is a common strategy for tackling many calculus problems. By understanding and practicing these techniques, you can confidently approach derivative problems and apply them in various contexts. The derivative is a powerful tool for understanding rates of change and modeling real-world phenomena, making its mastery essential for anyone pursuing studies or careers in mathematics, science, engineering, or related fields.

Complete the table to find the derivative of the function.