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To solve this problem, we will differentiate each function as requested. Here are the solutions for parts (a), (b), and (c).

### Given functions:
1. \( f(x) = \cos(2x) \)
2. \( g(x) = \ln(3x - 5) \)

### Solution Steps:

---

#### Part (a): Find \( f'(x) \)

To differentiate \( f(x) = \cos(2x) \), we apply the chain rule. 

\[
f'(x) = \frac{d}{dx} \cos(2x) = -\sin(2x) \cdot \frac{d}{dx}(2x)
\]

Since \( \frac{d}{dx}(2x) = 2 \), we have:

\[
f'(x) = -2 \sin(2x)
\]

---

#### Part (b): Find \( g'(x) \)

For \( g(x) = \ln(3x - 5) \), we apply the chain rule again.

\[
g'(x) = \frac{d}{dx} \ln(3x - 5) = \frac{1}{3x - 5} \cdot \frac{d}{dx}(3x - 5)
\]

Since \( \frac{d}{dx}(3x - 5) = 3 \), we have:

\[
g'(x) = \frac{3}{3x - 5}
\]

---

#### Part (c): Let \( h(x) = f(x) \cdot g(x) \). Find \( h'(x) \)

For \( h(x) = f(x) \cdot g(x) = \cos(2x) \cdot \ln(3x - 5) \), we use the product rule:

\[
h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\]

Substitute \( f'(x) = -2 \sin(2x) \) and \( g(x) = \ln(3x - 5) \):

\[
h'(x) = (-2 \sin(2x)) \cdot \ln(3x - 5) + \cos(2x) \cdot \frac{3}{3x - 5}
\]

Simplifying, we get:

\[
h'(x) = -2 \sin(2x) \ln(3x - 5) + \frac{3 \cos(2x)}{3x - 5}
\]

---

### Summary of Results:

1. \( f'(x) = -2 \sin(2x) \)
2. \( g'(x) = \frac{3}{3x - 5} \)
3. \( h'(x) = -2 \sin(2x) \ln(3x - 5) + \frac{3 \cos(2x)}{3x - 5} \)

Would you like further details on any part, or have any additional questions?

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### Related Questions:

1. How would you find \( f''(x) \) for \( f(x) = \cos(2x) \)?
2. What is the derivative of \( h(x) = f(x) \cdot g(x) \) if \( f(x) \) and \( g(x) \) are both exponential functions?
3. Can you use the product rule to find the derivative of \( h(x) = x \sin(x) \ln(x) \)?
4. How do we find \( h'(x) \) if \( h(x) = f(x) + g(x) \)?
5. How does the chain rule work when differentiating composite functions like \( f(g(x)) \)?

### Tip:
When differentiating composite functions, always identify the "inner" and "outer" functions to apply the chain rule accurately.

Let f(x) = cos(2x) and g(x) = ln(3x - 5). (a) Find f'(x). (b) Find g'(x). (c) Let h(x) = f(x) * g(x). Find h'(x).

This solution provides a step-by-step approach to differentiate f(x) = cos(2x) and g(x) = ln(3x - 5) using the chain and product rules. Learn how to apply these rules to find f'(x), g'(x), and h'(x) where h(x) = f(x) * g(x). Suitable for students in Grades 11-12.

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