https://file.aiquickdraw.com/mathbot/file/17261043916593ymgj3yq.jpg

To solve the problem, we need to find the derivative of the given function \( y = \frac{\pi}{5} \sin(\theta) - \sin(\theta) \) with respect to \(\theta\).

### Step-by-Step Solution:

1. **Differentiate each term separately:**

   - The derivative of \( \frac{\pi}{5} \sin(\theta) \) with respect to \(\theta\):
     \[
     \frac{d}{d\theta} \left( \frac{\pi}{5} \sin(\theta) \right) = \frac{\pi}{5} \cos(\theta)
     \]
   - The derivative of \( -\sin(\theta) \) with respect to \(\theta\):
     \[
     \frac{d}{d\theta} (-\sin(\theta)) = -\cos(\theta)
     \]

2. **Combine the results:**
   \[
   y' = \frac{\pi}{5} \cos(\theta) - \cos(\theta)
   \]

3. **Factor out the common term \( \cos(\theta) \):**
   \[
   y' = \cos(\theta) \left(\frac{\pi}{5} - 1\right)
   \]

### Final Answer:
\[
y' = \left(\frac{\pi}{5} - 1\right) \cos(\theta)
\]

Would you like any further details on this solution? Here are some related questions:

1. What is the derivative of \( \sin(\theta) \) with respect to \( \theta \)?
2. How do you factor common terms in a mathematical expression?
3. What is the significance of the cosine function in trigonometric derivatives?
4. How does the constant factor affect the differentiation process?
5. Can you explain the chain rule in differentiation?

**Tip:** When differentiating trigonometric functions, remember that the derivative of \( \sin(\theta) \) is \( \cos(\theta) \) and the derivative of \( \cos(\theta) \) is \( -\sin(\theta) \).

Use the rules of differentiation to find the derivative of the function: y = (π/5)sin(θ) - sin(θ).

Learn how to find the derivative of the function y = (π/5)sin(θ) - sin(θ) using basic rules of differentiation. This problem covers key concepts from calculus, including trigonometric differentiation and factoring common terms. Ideal for students in Grade 11-12 or introductory calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation