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To find the derivative \( f'(x) \) of the function \( f(x) = 10 \sin(4x) + 3 \cos(2x) \), we'll differentiate each term separately.

### Step 1: Differentiate \( 10 \sin(4x) \)
- The derivative of \( \sin(4x) \) with respect to \( x \) is:
  \[
  \frac{d}{dx} \sin(4x) = 4 \cos(4x)
  \]
- Therefore, the derivative of \( 10 \sin(4x) \) is:
  \[
  \frac{d}{dx} \left( 10 \sin(4x) \right) = 10 \times 4 \cos(4x) = 40 \cos(4x)
  \]

### Step 2: Differentiate \( 3 \cos(2x) \)
- The derivative of \( \cos(2x) \) with respect to \( x \) is:
  \[
  \frac{d}{dx} \cos(2x) = -2 \sin(2x)
  \]
- Therefore, the derivative of \( 3 \cos(2x) \) is:
  \[
  \frac{d}{dx} \left( 3 \cos(2x) \right) = 3 \times (-2) \sin(2x) = -6 \sin(2x)
  \]

### Step 3: Combine the results
- Adding the derivatives from Step 1 and Step 2:
  \[
  f'(x) = 40 \cos(4x) - 6 \sin(2x)
  \]

So the derivative \( f'(x) \) is:
\[
f'(x) = 40 \cos(4x) - 6 \sin(2x)
\]

Would you like any more details or have any questions?

Here are 5 related questions to explore:
1. What is the derivative of \( f(x) = \sin(5x) + 7 \cos(3x) \)?
2. How would you find the second derivative \( f''(x) \) for \( f(x) = 10 \sin(4x) + 3 \cos(2x) \)?
3. How does the derivative of \( \sin(kx) \) generally behave when \( k \) is a constant?
4. Can you apply the product rule to differentiate a function like \( f(x) = x^2 \sin(3x) \)?
5. What is the integral of \( f'(x) \) you just found?

**Tip:** When differentiating trigonometric functions involving coefficients (like \( \sin(kx) \) or \( \cos(kx) \)), remember to apply the chain rule to account for the coefficient of \( x \).

Given f(x) = 10 sin(4x) + 3 cos(2x), find f'(x).

Learn how to find the derivative of f(x) = 10 sin(4x) + 3 cos(2x) using the chain rule. This problem covers key calculus concepts such as differentiation of trigonometric functions and applying the chain rule. Suitable for high school students in Grades 10-12.

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