Derivative Of Sin X 3

Unveiling the Mystery: Deriving the Derivative of sin(x³), Step-by-Step

Finding the derivative of trigonometric functions often presents a unique challenge for students learning calculus. This article delves into the process of deriving the derivative of sin(x³), a seemingly complex function that elegantly unravels with the application of the chain rule. We will not only show you how to derive it, but also why each step is necessary, building a strong foundation in understanding the underlying principles of calculus. By the end, you'll be equipped to tackle similar problems with confidence.

Introduction

The core concept we'll be leveraging is the chain rule. This powerful rule allows us to differentiate composite functions – functions within functions. In our case, sin(x³) is a composite function: the sine function is applied to the function x³. Understanding the chain rule is crucial for mastering differentiation in calculus.

The Chain Rule: A Quick Refresher

The chain rule states that the derivative of a composite function, f(g(x)), is given by:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

In simpler terms: The derivative of the "outside" function (evaluated at the "inside" function) multiplied by the derivative of the "inside" function.

Step-by-Step Derivation of d/dx [sin(x³)]

Let's break down the derivation into manageable steps:

1. Identifying the Outer and Inner Functions:

   Our composite function is sin(x³).

   • The outer function is f(u) = sin(u), where 'u' represents the inner function.
   • The inner function is g(x) = x³.

2. Finding the Derivative of the Outer Function:

   The derivative of sin(u) with respect to u is: f'(u) = cos(u)

3. Finding the Derivative of the Inner Function:

   The derivative of x³ with respect to x is: g'(x) = 3x²

4. Applying the Chain Rule:

   Now, we combine the derivatives using the chain rule formula:

   d/dx [sin(x³)] = f'(g(x)) * g'(x) = cos(x³) * 3x²

5. Simplifying the Result:

   The final, simplified derivative of sin(x³) is:

   d/dx [sin(x³)] = 3x²cos(x³)

Explanation of Each Step: A Deeper Dive

Let's analyze each step in more detail to solidify your understanding.

• Step 1: Function Decomposition: Identifying the outer and inner functions is the crucial first step. Correctly separating the composite function is essential for accurate application of the chain rule. Practice this step thoroughly; it's fundamental to mastering derivative calculations.

• Step 2: Derivative of the Outer Function: This step utilizes the standard derivative of the sine function. Remember that the derivative of sin(u) with respect to u is cos(u). This is a foundational trigonometric derivative that you should memorize.

• Step 3: Derivative of the Inner Function: This step employs the power rule of differentiation. The power rule states that the derivative of xⁿ is nxⁿ⁻¹. Therefore, the derivative of x³ is 3x².

• Step 4: The Power of the Chain Rule: This is where the magic happens. The chain rule connects the derivatives of the outer and inner functions, providing a systematic way to differentiate composite functions. Understanding the logic behind this step – multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function – is key to mastering differentiation techniques.

• Step 5: Simplification: While this may seem like a trivial step, it's important for presenting your answer in a clear and concise manner. Always aim to simplify your derivative to its most compact form.

Illustrative Example: Evaluating the Derivative at a Specific Point

Let's say we want to find the derivative of sin(x³) at x = 1. We simply substitute x = 1 into our derived equation:

d/dx [sin(x³)] = 3x²cos(x³)

At x = 1:

3(1)²cos(1³) = 3cos(1)

This gives us the instantaneous rate of change of sin(x³) at x = 1. The exact value requires a calculator, but the process illustrates how to use the derived equation to find the derivative at a specific point.

Expanding Your Understanding: Related Derivatives

The process demonstrated above can be extended to other trigonometric functions and composite functions in general. For instance, let's consider deriving the derivative of cos(x⁴):

1. Outer function: cos(u)
2. Inner function: u = x⁴
3. Derivative of outer function: -sin(u)
4. Derivative of inner function: 4x³
5. Applying the chain rule: d/dx[cos(x⁴)] = -sin(x⁴) * 4x³ = -4x³sin(x⁴)

This exemplifies the versatility and broad applicability of the chain rule in differentiating composite functions.

Frequently Asked Questions (FAQ)

• Q: What if the inner function is more complex?

  A: The process remains the same. Just apply the chain rule methodically, breaking down the composite function into its constituent parts and applying the appropriate derivative rules to each part. For example, if the inner function was something like (2x + 1)³, you would first differentiate the outer function (cubed) and then the inner function (2x + 1) using the chain rule and the power rule, respectively.

• Q: Are there any limitations to the chain rule?

  A: The chain rule is very powerful but requires differentiability of both the inner and outer functions. If either function isn't differentiable at a specific point, the chain rule cannot be directly applied at that point.

• Q: How can I practice more?

  A: The best way to master this is through practice. Work through numerous examples with varying degrees of complexity. Start with simpler composite functions and gradually move towards more challenging ones. Online resources and textbooks offer a wealth of practice problems.

• Q: What if I encounter other trigonometric functions within a composite function?

  A: The method remains consistent. Remember the basic derivatives of trigonometric functions:

  • d/dx[sin(x)] = cos(x)
  • d/dx[cos(x)] = -sin(x)
  • d/dx[tan(x)] = sec²(x)
  • d/dx[cot(x)] = -csc²(x)
  • d/dx[sec(x)] = sec(x)tan(x)
  • d/dx[csc(x)] = -csc(x)cot(x)

  And apply the chain rule as needed.

Conclusion

Deriving the derivative of sin(x³) provides a valuable illustration of the chain rule's application. By methodically dissecting the composite function and applying the chain rule step-by-step, we arrive at the derivative: 3x²cos(x³). Remember that mastering calculus requires consistent practice and a thorough understanding of fundamental concepts like the chain rule. Use this detailed explanation as a foundation, practice diligently, and you will confidently navigate the world of derivatives. The key is not just to memorize the final answer but to deeply understand why that answer is correct. This comprehension will unlock your ability to tackle even more complex problems in the future.