Find *y* ′ and *y″*.

*y* = 

sin(*x*)

Given the function:

\[
y = \sin(x)
\]

To find the first derivative, \( y' \), and the second derivative, \( y'' \), we proceed as follows:

### Step 1: Find the first derivative (\( y' \))

The derivative of \( \sin(x) \) with respect to \( x \) is:

\[
y' = \cos(x)
\]

### Step 2: Find the second derivative (\( y'' \))

Now, take the derivative of \( y' = \cos(x) \). The derivative of \( \cos(x) \) is:

\[
y'' = -\sin(x)
\]

### Final Answer:
\[
y' = \cos(x)
\]
\[
y'' = -\sin(x)
\]

Would you like further clarification or details on these steps?

Here are 5 related questions for further practice:

1. What is the third derivative of \( y = \sin(x) \)?
2. What is the derivative of \( y = \cos(x) \)?
3. How would you differentiate \( y = \sin(2x) \)?
4. What is the second derivative of \( y = \sin^2(x) \)?
5. How do you find the derivative of \( y = e^x \sin(x) \)?

**Tip:** To find the second derivative of a function, simply differentiate the first derivative again.

Learn how to find the first and second derivatives of y = sin(x) with detailed steps. This calculus problem covers key concepts such as derivatives of trigonometric functions and their applications. Suitable for students in Grades 10-12.

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