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We are tasked with finding the derivative of the function:

\[
f(x) = \cosh(2 - 3x^2)
\]

### Step-by-step Solution:

1. **Chain Rule**: The function \( f(x) \) is a composition of functions, so we need to apply the chain rule. First, differentiate the outer function, \( \cosh(u) \), with respect to \( u \), then multiply by the derivative of the inner function \( u = 2 - 3x^2 \).

2. **Derivative of \( \cosh(u) \)**: The derivative of \( \cosh(u) \) is \( \sinh(u) \). Therefore, we get:
   \[
   \frac{d}{dx} \left( \cosh(2 - 3x^2) \right) = \sinh(2 - 3x^2) \cdot \frac{d}{dx}(2 - 3x^2)
   \]

3. **Derivative of \( 2 - 3x^2 \)**: The derivative of \( 2 - 3x^2 \) is:
   \[
   \frac{d}{dx}(2 - 3x^2) = -6x
   \]

4. **Final Expression**: Now multiply the derivative of the inner function by the derivative of the outer function:
   \[
   f'(x) = -6x \cdot \sinh(2 - 3x^2)
   \]

### Correct Answer:
The correct option is:
\[
\boxed{\text{b.} \, f'(x) = -6x \cdot \sinh(2 - 3x^2)}
\]

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Do you want more details on any of these steps? Here are five related questions for practice:

1. What is the derivative of \( \sinh(x) \)?
2. How do you differentiate the function \( \cosh(x^2) \)?
3. What is the chain rule and how is it applied?
4. Can you derive the derivative of a general hyperbolic function \( \cosh(u) \)?
5. How do you find the second derivative of the given function?

**Tip:** The derivative of hyperbolic functions, like \( \cosh \) and \( \sinh \), are closely related, just like trigonometric functions.

Find the derivative of the given function: f(x) = cosh(2 - 3x^2)

Learn how to differentiate the hyperbolic function f(x) = cosh(2 - 3x^2) using the chain rule. This solution involves key calculus concepts and hyperbolic function derivatives, making it suitable for college-level students.

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