`e^("sin x")`

Answer

To differentiate the function \( e^{\sin x} \), we will use the chain rule. 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**:
- Let \( u = \sin x \) (inner function).
- Then, the function can be rewritten as \( e^u \) (outer function).

2. **Differentiate the outer function**:
- The derivative of \( e^u \) with respect to \( u \) is \( e^u \).

3. **Differentiate the inner function**:
- The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).

4. **Apply the chain rule**:
- According to the chain rule, the derivative of \( e^{\sin x} \) is:
\[ \frac{d}{dx}(e^{\sin x}) = e^{\sin x} \cdot \frac{d}{dx}(\sin x) = e^{\sin x} \cdot \cos x \]

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

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