Differentiating (sin Θ - Cos 2θ) / (1 + Sin 2θ) A Step-by-Step Guide

\nDifferentiation is a fundamental concept in calculus that involves finding the rate of change of a function. In this comprehensive guide, we will walk through the process of differentiating the given function: (sin θ - cos 2θ) / (1 + sin 2θ). This problem combines trigonometric functions and the quotient rule, offering a rich exercise in calculus techniques. Let's break it down step by step, ensuring clarity and understanding at each stage.

Understanding the Problem

Before we dive into the solution, let's make sure we understand the problem clearly. We are asked to differentiate the function:

$$f(\theta) = \frac{\sin \theta - \cos 2\theta}{1 + \sin 2\theta}$$

This involves finding the derivative of f(θ) with respect to θ, denoted as f'(θ) or df/dθ. To tackle this, we'll need to employ several key calculus concepts, including:

• Trigonometric Derivatives: Knowing the derivatives of basic trigonometric functions like sin θ and cos θ.
• Double Angle Formulas: Using trigonometric identities to simplify expressions involving sin 2θ and cos 2θ.
• Quotient Rule: Applying the rule for differentiating a quotient of two functions.

Prerequisites

Before we proceed, let’s ensure we have a solid grasp of the necessary prerequisites. These include:

1. Derivatives of Trigonometric Functions:

   • The derivative of sin θ is cos θ.
   • The derivative of cos θ is -sin θ.

2. Double Angle Formulas:

   • cos 2θ = cos² θ - sin² θ
   • cos 2θ = 1 - 2sin² θ
   • cos 2θ = 2cos² θ - 1
   • sin 2θ = 2sin θ cos θ

3. Quotient Rule:

   • If we have a function f(θ) = g(θ) / h(θ), then its derivative f'(θ) is given by:

   $$f'(\theta) = \frac{g'(\theta)h(\theta) - g(\theta)h'(\theta)}{[h(\theta)]^2}$$

With these prerequisites in mind, we are well-equipped to tackle the problem at hand. Let's move on to the solution step by step.

Step-by-Step Solution

Step 1: Identify g(θ) and h(θ)

First, let’s identify the numerator and denominator of our function:

• g(θ) = sin θ - cos 2θ
• h(θ) = 1 + sin 2θ

Step 2: Find g'(θ) and h'(θ)

Next, we need to find the derivatives of g(θ) and h(θ) with respect to θ.

Finding g'(θ):

g(θ) = sin θ - cos 2θ

The derivative of sin θ is cos θ. For the derivative of cos 2θ, we use the chain rule. The derivative of cos u is -sin u, and the derivative of 2θ is 2. So,

$$\frac{d}{d\theta}(\cos 2\theta) = -\sin(2\theta) \cdot 2 = -2\sin 2\theta$$

Thus,

$$g'(\theta) = \cos \theta - (-2\sin 2\theta) = \cos \theta + 2\sin 2\theta$$

Finding h'(θ):

h(θ) = 1 + sin 2θ

The derivative of a constant (1) is 0. For the derivative of sin 2θ, we again use the chain rule. The derivative of sin u is cos u, and the derivative of 2θ is 2. So,

$$\frac{d}{d\theta}(\sin 2\theta) = \cos(2\theta) \cdot 2 = 2\cos 2\theta$$

Thus,

$$h'(\theta) = 0 + 2\cos 2\theta = 2\cos 2\theta$$

Step 3: Apply the Quotient Rule

Now that we have g(θ), h(θ), g'(θ), and h'(θ), we can apply the quotient rule:

$$f'(\theta) = \frac{g'(\theta)h(\theta) - g(\theta)h'(\theta)}{[h(\theta)]^2}$$

Plugging in our values:

$$f'(\theta) = \frac{(\cos \theta + 2\sin 2\theta)(1 + \sin 2\theta) - (\sin \theta - \cos 2\theta)(2\cos 2\theta)}{(1 + \sin 2\theta)^2}$$

Step 4: Expand and Simplify

Next, we expand the numerator and try to simplify the expression:

$$f'(\theta) = \frac{\cos \theta + \cos \theta \sin 2\theta + 2\sin 2\theta + 2\sin^2 2\theta - 2\sin \theta \cos 2\theta + 2\cos^2 2\theta}{(1 + \sin 2\theta)^2}$$

We can rearrange terms and use the identity sin² x + cos² x = 1 to simplify:

$$f'(\theta) = \frac{\cos \theta + \cos \theta \sin 2\theta + 2\sin 2\theta - 2\sin \theta \cos 2\theta + 2(\sin^2 2\theta + \cos^2 2\theta)}{(1 + \sin 2\theta)^2}$$

$$f'(\theta) = \frac{\cos \theta + \cos \theta \sin 2\theta + 2\sin 2\theta - 2\sin \theta \cos 2\theta + 2}{(1 + \sin 2\theta)^2}$$

Step 5: Use Double Angle Formulas to Simplify Further

Recall the double angle formulas:

• sin 2θ = 2sin θ cos θ
• cos 2θ = 1 - 2sin² θ

Substitute sin 2θ in the expression:

$$f'(\theta) = \frac{\cos \theta + \cos \theta (2\sin \theta \cos \theta) + 2(2\sin \theta \cos \theta) - 2\sin \theta \cos 2\theta + 2}{(1 + 2\sin \theta \cos \theta)^2}$$

$$f'(\theta) = \frac{\cos \theta + 2\sin \theta \cos^2 \theta + 4\sin \theta \cos \theta - 2\sin \theta \cos 2\theta + 2}{(1 + 2\sin \theta \cos \theta)^2}$$

Now substitute cos 2θ = 1 - 2sin² θ:

$$f'(\theta) = \frac{\cos \theta + 2\sin \theta \cos^2 \theta + 4\sin \theta \cos \theta - 2\sin \theta (1 - 2\sin^2 \theta) + 2}{(1 + 2\sin \theta \cos \theta)^2}$$

$$f'(\theta) = \frac{\cos \theta + 2\sin \theta \cos^2 \theta + 4\sin \theta \cos \theta - 2\sin \theta + 4\sin^3 \theta + 2}{(1 + 2\sin \theta \cos \theta)^2}$$

Step 6: Look for Further Simplifications

At this point, the expression looks quite complex, but we can try to rearrange terms and look for possible factorizations or simplifications. The denominator is (1 + sin 2θ)², which can also be written as (1 + 2sin θ cos θ)². The numerator is:

$$\cos \theta + 2\sin \theta \cos^2 \theta + 4\sin \theta \cos \theta - 2\sin \theta + 4\sin^3 \theta + 2$$

Let's rearrange it:

$$4\sin^3 \theta + 2\sin \theta \cos^2 \theta + 4\sin \theta \cos \theta - 2\sin \theta + \cos \theta + 2$$

This is a complex expression, and further simplification might not be straightforward. However, let’s try to factor or combine terms. We know that cos² θ = 1 - sin² θ, so we can rewrite the expression as:

$$4\sin^3 \theta + 2\sin \theta (1 - \sin^2 \theta) + 4\sin \theta \cos \theta - 2\sin \theta + \cos \theta + 2$$

$$4\sin^3 \theta + 2\sin \theta - 2\sin^3 \theta + 4\sin \theta \cos \theta - 2\sin \theta + \cos \theta + 2$$

$$2\sin^3 \theta + 4\sin \theta \cos \theta + \cos \theta + 2$$

Now our expression for f'(θ) is:

$$f'(\theta) = \frac{2\sin^3 \theta + 4\sin \theta \cos \theta + \cos \theta + 2}{(1 + 2\sin \theta \cos \theta)^2}$$

Step 7: Final Simplification (If Possible)

We can rewrite the numerator as:

$$2\sin^3 \theta + 2 + 4\sin \theta \cos \theta + \cos \theta$$

And the denominator is:

$$(1 + 2\sin \theta \cos \theta)^2 = (1 + \sin 2\theta)^2$$

The numerator doesn't appear to simplify easily, so let's stick with our expression:

$$f'(\theta) = \frac{2\sin^3 \theta + 4\sin \theta \cos \theta + \cos \theta + 2}{(1 + \sin 2\theta)^2}$$

This may be as simplified as we can get it without additional trigonometric identities or manipulations.

Final Answer

After applying the quotient rule, trigonometric derivatives, and double angle formulas, we found the derivative of the given function to be:

$$f'(\theta) = \frac{2\sin^3 \theta + 4\sin \theta \cos \theta + \cos \theta + 2}{(1 + \sin 2\theta)^2}$$

This comprehensive step-by-step solution should help anyone understand the process of differentiating complex trigonometric functions. Remember, the key is to break the problem down into manageable steps, apply the appropriate rules, and simplify as much as possible.

Common Mistakes to Avoid

When differentiating trigonometric functions, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and improve your accuracy. Here are some of the most frequent errors:

1. Incorrect Application of the Chain Rule: The chain rule is crucial when differentiating composite functions, such as sin(2θ) or cos(3θ). A common mistake is forgetting to multiply by the derivative of the inner function. For example, the derivative of sin(2θ) is 2cos(2θ), not just cos(2θ). Always remember to account for the derivative of the inner function.

2. Misremembering Trigonometric Derivatives: It’s essential to know the derivatives of basic trigonometric functions. The derivative of sin θ is cos θ, and the derivative of cos θ is -sin θ. Forgetting the negative sign in the derivative of cos θ is a common mistake. Ensure you have these derivatives memorized correctly.

3. Errors with the Quotient Rule: The quotient rule can be tricky if not applied carefully. The correct formula is:

   $$(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$$

   A frequent error is mixing up the order of terms in the numerator or forgetting to square the denominator. Double-check your application of the quotient rule to avoid these mistakes.

4. Incorrect Use of Trigonometric Identities: Trigonometric identities are powerful tools for simplifying expressions before or after differentiation. However, using them incorrectly can lead to significant errors. For example, misapplying the double angle formulas or Pythagorean identities can complicate the problem. Always ensure you are using the correct identity and applying it appropriately.

5. Algebraic Simplification Errors: After applying differentiation rules, simplifying the resulting expression is crucial. Mistakes in algebraic manipulation, such as incorrectly expanding terms, combining like terms, or factoring, can lead to a wrong final answer. Take your time and double-check each step of the simplification process.

6. Forgetting the Constant of Integration: While this mistake is more relevant in integration, it's worth mentioning for completeness. When finding indefinite integrals, always add the constant of integration (C) because the derivative of a constant is zero. In differentiation problems, however, there is no need to add a constant.

7. Not Simplifying Before Differentiating: Sometimes, simplifying the original function before differentiating can make the process much easier. For example, using trigonometric identities to rewrite the function in a simpler form can reduce the complexity of the derivative. Always look for opportunities to simplify before diving into differentiation.

8. Ignoring the Domain of the Function: Be mindful of the domain of the function, especially when dealing with trigonometric functions. Certain values of θ may make the function undefined, and this should be considered when interpreting the results. For example, denominators cannot be zero, so you need to exclude values that make the denominator zero.

By being aware of these common mistakes and taking extra care in your calculations, you can improve your accuracy and confidence in differentiating trigonometric functions. Remember to practice regularly and review your work to catch errors.

Practice Problems

To solidify your understanding of differentiating trigonometric functions, working through practice problems is essential. Here are a few problems you can try, covering various aspects of the topic. Attempt these problems on your own, and then review the solutions to check your work and understanding.

1. Differentiate f(x) = sin(3x) + cos(2x):

   This problem involves applying the chain rule to differentiate composite trigonometric functions. Remember to multiply by the derivative of the inner function.

2. Find the derivative of g(x) = x² sin(x):

   Here, you’ll need to use the product rule, which states that (uv)' = u'v + uv'. Be careful to differentiate both x² and sin(x) correctly.

3. Differentiate h(x) = (sin x) / (1 + cos x):

   This problem requires the quotient rule: (u/v)' = (u'v - uv') / v². Make sure to apply the rule accurately and simplify the resulting expression.

4. Find the derivative of k(x) = tan(x) = sin(x) / cos(x):

   You can differentiate tan(x) directly using its derivative, or you can use the quotient rule on sin(x) / cos(x). This exercise reinforces the application of the quotient rule and the derivatives of sin(x) and cos(x).

5. Differentiate l(x) = cos²(x):

   This problem can be approached using the chain rule. Rewrite cos²(x) as (cos(x))² and apply the power rule in conjunction with the chain rule.

6. Find the derivative of m(x) = sin(2x) cos(x):

   This requires the product rule and the chain rule. Remember to use the double angle formula for sin(2x) if needed to simplify the expression.

7. Differentiate n(x) = √(sin(x)):

   Rewrite the function as (sin(x))^(1/2) and apply the chain rule along with the power rule. This problem tests your understanding of differentiating composite functions involving square roots.

By working through these practice problems, you’ll gain confidence in your ability to differentiate trigonometric functions. Remember to review your work, identify any mistakes, and understand the correct solutions. Practice is key to mastering calculus concepts.

Conclusion

Differentiating the function (sin θ - cos 2θ) / (1 + sin 2θ) is a complex yet rewarding exercise in calculus. By meticulously applying the quotient rule, trigonometric derivatives, and double angle formulas, we can arrive at the derivative. It’s crucial to practice these techniques to build confidence and accuracy. Remember, calculus is a journey of continuous learning and refinement. Keep practicing, and you’ll master these concepts in no time!