https://file.aiquickdraw.com/mathbot/file/1729126063812517bz7l5.jpg

We are given the function \( f(x) = \sin(x^2) \) and asked to find its derivative \( f'(x) \).

To solve this, we will apply the chain rule. The chain rule is used when differentiating composite functions, and it states:

\[
\frac{d}{dx} \left[ \sin(g(x)) \right] = \cos(g(x)) \cdot g'(x)
\]

### Step-by-step solution:

1. Start with \( f(x) = \sin(x^2) \).
2. Apply the chain rule:
   - The outer function is \( \sin(u) \), where \( u = x^2 \).
   - The derivative of \( \sin(u) \) is \( \cos(u) \), so we get \( \cos(x^2) \).
   - Now, multiply by the derivative of the inner function \( x^2 \), which is \( 2x \).

Thus, the derivative is:
\[
f'(x) = 2x \cos(x^2)
\]

Would you like further clarification or details on this? Here are 5 related questions for deeper exploration:

1. How is the chain rule applied to other trigonometric functions?
2. What is the derivative of \( f(x) = \cos(x^2) \)?
3. How would you differentiate \( f(x) = \sin(x^3) \)?
4. What happens when differentiating \( f(x) = \sin(x^2 + 1) \)?
5. Can you explain the second derivative of \( f(x) = \sin(x^2) \)?

**Tip:** When differentiating composite functions, always identify the outer and inner functions to apply the chain rule correctly.

Learn how to differentiate f(x) = sin(x^2) step-by-step using the chain rule. This problem covers key calculus concepts including trigonometric differentiation and composite functions. 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