https://file.aiquickdraw.com/mathbot/file/1727211549080twz1u3cd.pdf

For **Question 2, Part C** of the midterm:

We are tasked with calculating the following derivative:

\[
\frac{d}{dx} \left( 2 \sin (\pi k(x)) \right) \quad \text{at} \quad x = 4
\]

To solve this:

### Step-by-step process:
1. **Apply the Chain Rule**:  
   \[
   \frac{d}{dx} \left( 2 \sin (\pi k(x)) \right) = 2 \cdot \cos (\pi k(x)) \cdot \frac{d}{dx} (\pi k(x)) 
   \]
   Using the chain rule, we first take the derivative of the sine function and then the derivative of \(\pi k(x)\).

2. **Differentiate \(k(x)\)**:  
   \[
   \frac{d}{dx} (\pi k(x)) = \pi k'(x)
   \]

3. **Substitute \(x = 4\), and use given values from the table**:
   From the table, at \(x = 4\), we know:
   \[
   k(4) = 3 \quad \text{and} \quad k'(4) = -6
   \]

   Substituting these into our derivative expression:
   \[
   \frac{d}{dx} \left( 2 \sin (\pi k(x)) \right) \bigg|_{x=4} = 2 \cdot \cos (\pi \cdot 3) \cdot (\pi \cdot -6)
   \]

4. **Simplify**:
   \[
   \cos (\pi \cdot 3) = \cos(3\pi) = -1
   \]
   Substituting this into the equation:
   \[
   = 2 \cdot (-1) \cdot (\pi \cdot -6)
   \]
   \[
   = 12\pi
   \]

Thus, the value of the derivative at \(x = 4\) is **\(12\pi\)**.

---

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:
1. What is the Chain Rule and when is it applied?
2. How do you calculate the derivative of a composite function?
3. What is the significance of the value of \(k(x)\) and \(k'(x)\) in this problem?
4. How can trigonometric identities like \(\cos(\pi n)\) help simplify calculations?
5. What would the derivative look like for a different trigonometric function, like cosine?

**Tip:** Always remember to use the chain rule when differentiating a function within another function, like \(\sin(\pi k(x))\).

Learn how to compute the derivative of 2sin(πk(x)) at x=4 using the chain rule and trigonometric identities. This detailed solution explores key calculus concepts and provides a step-by-step process, ideal for college-level calculus students.

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