What is the derivation of `sin(sinx)` ?

Options

A. `cos(cosx)`
B. `cos(sinx)`
C. `cos(sinx)cosx`
D. `cos(cosx)cosx`

Answer

To find the derivative of \( \sin(\sin x) \), 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 becomes \( f(u) = \sin(u) \) (outer function).

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

3. **Differentiate the inner function**:
- The derivative of \( u = \sin x \) is \( \frac{du}{dx} = \cos x \).

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

5. **Final Result**:
- Therefore, the derivative of \( \sin(\sin x) \) is:
\[ \frac{d}{dx} \sin(\sin x) = \cos(\sin x) \cdot \cos x \]