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We are given that \( f(x) = \sin(x) \). Now let's find the second, third, and fourth derivatives step by step:

### Step 1: First derivative
We know that:
\[
f'(x) = \frac{d}{dx} \left( \sin(x) \right) = \cos(x)
\]

### Step 2: Second derivative
The second derivative is the derivative of \( f'(x) \), so:
\[
f''(x) = \frac{d}{dx} \left( \cos(x) \right) = -\sin(x)
\]

### Step 3: Third derivative
The third derivative is the derivative of \( f''(x) \), so:
\[
f'''(x) = \frac{d}{dx} \left( -\sin(x) \right) = -\cos(x)
\]

### Step 4: Fourth derivative
The fourth derivative is the derivative of \( f'''(x) \), so:
\[
f''''(x) = \frac{d}{dx} \left( -\cos(x) \right) = \sin(x)
\]

Thus, the derivatives are:
- \( f''(x) = -\sin(x) \)
- \( f'''(x) = -\cos(x) \)
- \( f''''(x) = \sin(x) \)

Do you have any questions or need more details?

Here are five related questions to deepen your understanding:
1. What is the fifth derivative of \( \sin(x) \)?
2. Can you find the second derivative of \( f(x) = \cos(x) \)?
3. How does the periodic nature of sine and cosine reflect in higher-order derivatives?
4. If \( f(x) = \sin(2x) \), what would the second derivative be?
5. How do derivatives of trigonometric functions change when a coefficient is inside the sine or cosine function?

**Tip:** Trigonometric functions, like sine and cosine, have a cyclical pattern in their derivatives, repeating every four derivatives!

Let f(x) = sin(x), find each of the following: Find the second derivative, f''(x), Find the third derivative, f'''(x), Find the fourth derivative, f''''(x)

Learn how to find the second, third, and fourth derivatives of f(x) = sin(x) using basic calculus. Understand the cyclical pattern of sine and cosine derivatives. This problem is suitable for students in Grades 11-12.

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