Derivative Of -2sin²(3x⁴) Using Chain Rule A Step-by-Step Guide

Introduction

In the realm of calculus, finding derivatives is a fundamental operation. It allows us to understand the rate at which a function changes. In this article, we will embark on a journey to find the derivative of the function $-2 \sin^2(3x^4)$. This problem is a classic example that requires the application of the chain rule, a powerful tool in calculus for differentiating composite functions. Our discussion will navigate through each step meticulously, ensuring a clear and comprehensive understanding of the solution. To truly master calculus, one must not only grasp the formulas but also the underlying concepts that drive them. As we delve into this problem, we will emphasize the conceptual understanding alongside the procedural steps. We will begin by revisiting the chain rule, which will be the cornerstone of our approach. Then, we will systematically dismantle the given function, identifying the layers of composition and applying the chain rule to each layer. Through careful application of the power rule, trigonometric derivatives, and algebraic manipulations, we will arrive at the final derivative. This exploration is more than just a mathematical exercise; it's a journey into the heart of calculus, where we discover the elegance and power of differentiation.

Revisiting the Chain Rule

The chain rule is a cornerstone of differential calculus, providing a method for computing the derivative of a composite function. A composite function is essentially a function within a function, like peeling layers of an onion. The chain rule states that if we have a composite function $y = f(g(x))$, where $f$ and $g$ are differentiable functions, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$$

In simpler terms, we differentiate the outer function $f$ with respect to the inner function $g(x)$, and then multiply it by the derivative of the inner function $g(x)$ with respect to $x$. This rule is crucial when dealing with functions that are composed of multiple layers, as it allows us to break down the differentiation process into manageable steps. To illustrate the chain rule further, consider an analogy: Imagine you're turning on a light in a room. The action involves multiple steps: flipping the switch (inner function) and the light turning on (outer function). The chain rule helps us understand how the overall rate of change (light turning on) is affected by the rate of change of each step (flipping the switch). We'll use this concept extensively in solving our problem, where we have a function composed of trigonometric, power, and polynomial functions. Mastering the chain rule is not just about memorizing the formula; it's about understanding the process of unraveling the layers of a composite function and differentiating each part systematically. As we proceed, we'll see how the chain rule becomes our guiding principle in navigating the complexities of our given function.

Deconstructing the Function: -2sin²(3x⁴)

Before we dive into the differentiation process, it's crucial to deconstruct the given function, $-2 \sin^2(3x^4)$, and identify its layers of composition. This will guide us in applying the chain rule effectively. The function can be viewed as a composition of several functions, each nested within the other. We can break it down as follows:

1. Outermost function: $-2u^2$, where $u = \sin(3x^4)$. This is a power function multiplied by a constant.
2. Middle function: $u = \sin(v)$, where $v = 3x^4$. This is a trigonometric function.
3. Innermost function: $v = 3x^4$. This is a polynomial function.

Understanding this layered structure is key to applying the chain rule successfully. We will differentiate the outermost function first, then move inwards, differentiating each layer while keeping track of the derivatives at each step. Think of it like peeling an onion, where each layer reveals another function to differentiate. By identifying these layers, we create a roadmap for our differentiation journey. This methodical approach ensures that we don't miss any steps and that we apply the chain rule correctly at each stage. Furthermore, this decomposition helps us appreciate the complexity of even seemingly simple functions and the power of the chain rule in handling them. As we proceed, we'll see how this initial deconstruction lays the foundation for a clear and concise solution.

Applying the Chain Rule: Step-by-Step Differentiation

Now that we have deconstructed the function, we can apply the chain rule step-by-step to find the derivative of $-2 \sin^2(3x^4)$.

1. Differentiating the outermost function:

   We begin by differentiating $-2u^2$ with respect to $u$, where $u = \sin(3x^4)$. Using the power rule, we get:

   $$\frac{d}{du}(-2u^2) = -4u$$

   Substituting back $u = \sin(3x^4)$, we have:

   $$-4 \sin(3x^4)$$

2. Differentiating the middle function:

   Next, we differentiate $u = \sin(v)$ with respect to $v$, where $v = 3x^4$. The derivative of $\sin(v)$ is $\cos(v)$, so we get:

   $$\frac{d}{dv}(\sin(v)) = \cos(v)$$

   Substituting back $v = 3x^4$, we have:

   $$\cos(3x^4)$$

3. Differentiating the innermost function:

   Finally, we differentiate $v = 3x^4$ with respect to $x$. Again using the power rule, we get:

   $$\frac{d}{dx}(3x^4) = 12x^3$$

4. Combining the Derivatives:

   Now, we multiply the derivatives obtained in each step according to the chain rule:

   $$\frac{d}{dx}(-2 \sin^2(3x^4)) = -4 \sin(3x^4) \cdot \cos(3x^4) \cdot 12x^3$$

   Simplifying, we get:

   $$-48x^3 \sin(3x^4) \cos(3x^4)$$

This step-by-step approach highlights the power of the chain rule in breaking down a complex differentiation problem into smaller, manageable parts. By systematically differentiating each layer of the function and then combining the results, we arrive at the final derivative. This process demonstrates not only the mechanics of the chain rule but also the importance of a methodical and organized approach in calculus. As we move towards the final simplification, we'll see how algebraic manipulations can further refine our result.

Simplifying the Result: Trigonometric Identities

Our derivative so far is $-48x^3 \sin(3x^4) \cos(3x^4)$. We can further simplify this expression using trigonometric identities. Recall the double-angle identity for sine:

$$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$

We can rewrite our derivative to take advantage of this identity. Notice that we have $\sin(3x^4) \cos(3x^4)$ in our expression. Multiplying and dividing by 2, we get:

$$-24x^3 \cdot 2 \sin(3x^4) \cos(3x^4)$$

Now, we can apply the double-angle identity with $\theta = 3x^4$:

$$-24x^3 \sin(2 \cdot 3x^4)$$

$$-24x^3 \sin(6x^4)$$

This simplification not only makes the expression more concise but also reveals the underlying structure of the derivative. By recognizing and applying trigonometric identities, we can often transform complex expressions into simpler forms. This step underscores the importance of having a strong foundation in trigonometry when working with calculus problems involving trigonometric functions. The final simplified derivative, $-24x^3 \sin(6x^4)$, is both elegant and informative, providing a clear representation of the rate of change of our original function.

Final Answer: The Derivative of -2sin²(3x⁴)

After meticulously applying the chain rule and simplifying using trigonometric identities, we have arrived at the final answer. The derivative of the function $-2 \sin^2(3x^4)$ is:

$$\frac{d}{dx}(-2 \sin^2(3x^4)) = -24x^3 \sin(6x^4)$$

This result encapsulates the entire process, from the initial deconstruction of the function to the final simplified expression. It demonstrates the power of the chain rule in handling composite functions and the importance of trigonometric identities in simplifying complex expressions. The journey to find this derivative has been a testament to the elegance and interconnectedness of calculus and trigonometry. Each step, from identifying the layers of the function to applying the chain rule and simplifying with identities, has contributed to a deeper understanding of the underlying principles. This final answer is not just a mathematical solution; it's a culmination of concepts and techniques, a testament to the beauty and power of mathematical reasoning. As we conclude this exploration, we carry with us not only the answer but also a reinforced appreciation for the tools and techniques of calculus.

Conclusion

In conclusion, finding the derivative of $-2 \sin^2(3x^4)$ has been a comprehensive exercise in applying the chain rule and trigonometric identities. We started by deconstructing the function into its composite layers, recognizing the nested nature of trigonometric and polynomial functions. This allowed us to systematically apply the chain rule, differentiating each layer while working from the outside in. The power rule, trigonometric derivatives, and algebraic manipulations were all essential tools in this process. Furthermore, we demonstrated the importance of simplification, using trigonometric identities to express the final derivative in a more concise and elegant form. The final result, $-24x^3 \sin(6x^4)$, not only answers the question but also showcases the interconnectedness of calculus and trigonometry. This exercise has reinforced the importance of a methodical approach in calculus, where each step builds upon the previous one to arrive at the solution. It has also highlighted the significance of understanding the underlying principles, such as the chain rule, rather than simply memorizing formulas. The journey through this problem has been a valuable learning experience, providing a deeper appreciation for the beauty and power of calculus. The ability to find derivatives is a fundamental skill in mathematics and its applications, and this exploration has provided a solid foundation for tackling more complex problems in the future. As we move forward, the lessons learned here will serve as a guide, reminding us of the importance of precision, patience, and a deep understanding of mathematical concepts.