To differentiate \( e^{\cos(3x)} \), we will apply the chain rule. Here's the step-by-step process:

Let:
\[ f(x) = e^{\cos(3x)} \]

### Step 1: Differentiate the outer function
The derivative of \( e^u \) with respect to \( u \) is \( e^u \), so the outer function differentiation gives:
\[
\frac{d}{dx}(e^{\cos(3x)}) = e^{\cos(3x)} \cdot \frac{d}{dx}(\cos(3x))
\]

### Step 2: Differentiate the inner function
Now, differentiate \( \cos(3x) \) with respect to \( x \). Using the chain rule again:
\[
\frac{d}{dx}(\cos(3x)) = -\sin(3x) \cdot 3
\]

### Step 3: Multiply the results
Now, multiply the result from Step 1 with the derivative from Step 2:
\[
\frac{d}{dx}(e^{\cos(3x)}) = e^{\cos(3x)} \cdot (-3\sin(3x))
\]

### Final Answer:
\[
\frac{d}{dx}(e^{\cos(3x)}) = -3e^{\cos(3x)}\sin(3x)
\]

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How would you differentiate \( e^{\sin(3x)} \)?
2. Can you differentiate \( \cos^2(3x) \)?
3. What is the second derivative of \( e^{\cos(3x)} \)?
4. How do you apply the chain rule to \( \tan(\ln(3x)) \)?
5. What is the integral of \( e^{\cos(3x)} \)?

**Tip:** When differentiating composite functions, always remember to apply the chain rule step by step for accuracy.

Learn step-by-step how to differentiate e^cos(3x) by applying the chain rule. This detailed guide covers the differentiation of exponential and trigonometric functions, and includes key formulas such as the derivative of e^u and cos(x). Suitable for advanced high school or college calculus students.

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