derivative of `"cos" e^(x)`

Answer

To find the derivative of the function \( f(x) = \cos(e^x) \), we will use the chain rule of differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Outer function: \( f(u) = \cos(u) \)
- Inner function: \( g(x) = e^x \)

2. **Differentiate the outer function**:
- The derivative of \( \cos(u) \) with respect to \( u \) is:
\[ f'(u) = -\sin(u) \]

3. **Differentiate the inner function**:
- The derivative of \( e^x \) with respect to \( x \) is:
\[ g'(x) = e^x \]

4. **Apply the chain rule**:
- According to the chain rule:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
- Substituting the derivatives we found:
\[ \frac{d}{dx} \cos(e^x) = -\sin(e^x) \cdot e^x \]

5. **Combine the results**:
- Therefore, the derivative of \( \cos(e^x) \) is:
\[ \frac{d}{dx} \cos(e^x) = -e^x \sin(e^x) \]

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