https://file.aiquickdraw.com/mathbot/file/1729247394619s5qol0k3.jpg

The function given is:

\[
y = \sin^2(x^2 + 4)
\]

We want to find the derivative \( y' \). To do this, we will use the chain rule.

### Steps:
1. Recognize that \( y = \sin^2(u) \), where \( u = x^2 + 4 \). We need to differentiate this expression.
2. Apply the chain rule:
   \[
   \frac{dy}{dx} = 2\sin(u) \cdot \frac{d}{dx}[\sin(u)] \cdot \frac{du}{dx}
   \]
3. Now, \( u = x^2 + 4 \), so \( \frac{du}{dx} = 2x \).
4. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), so:
   \[
   \frac{dy}{dx} = 2\sin(x^2 + 4) \cdot \cos(x^2 + 4) \cdot 2x
   \]
5. Simplifying this:
   \[
   y' = 4x\sin(x^2 + 4)\cos(x^2 + 4)
   \]

This can also be written using the double-angle identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \):
\[
y' = 2x\sin(2(x^2 + 4))
\]

### Final derivative:
\[
y' = 2x \sin(2(x^2 + 4))
\]

Would you like further explanation or have any questions?

---

Here are 5 questions related to this topic:

1. What is the chain rule and when do we apply it?
2. Can you explain how the double-angle identity works in trigonometric functions?
3. How does the derivative of \( \sin(u) \) behave when \( u \) is a function of \( x \)?
4. Why did we use the chain rule in this problem, and what would happen if we didn't?
5. What is the geometric meaning of the derivative of trigonometric functions?

**Tip**: When dealing with trigonometric functions raised to a power, always consider using both the chain rule and trigonometric identities for simplification!

Find the derivative of the function y = sin^2(x^2 + 4).

Learn how to find the derivative of y = sin^2(x^2 + 4) with a step-by-step explanation. This problem covers key calculus concepts, including the chain rule and trigonometric derivatives. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation