Question

If \(f(x) = \sin x e^{\sin x}\), find \(f'(x)\)

• A. \( \cos x e^{\sin x} + \sin x e^{\sin x} \)
• B. \( \cos x e^{\sin x} - \sin x e^{\sin x} \)
• C. \( \cos x e^{\sin x} + e^{\sin x} \)
• D. \( \cos x e^{\sin x} + \sin x e^{\cos x} \)

Hint:

When differentiating a product of functions, always use the product rule: \( (uv)' = u'v + uv' \).

Answer 1

The Correct Option is A

We need to differentiate the function \( f(x) = \sin x e^{\sin x} \). To do this, we will use the product rule for differentiation: \[ \frac{d}{dx} [u \cdot v] = u' \cdot v + u \cdot v' \] Here, let \( u = \sin x \) and \( v = e^{\sin x} \). Then: - \( u' = \cos x \), - \( v' = e^{\sin x} \cdot \cos x \) (using the chain rule). Now applying the product rule: \[ f'(x) = \cos x \cdot e^{\sin x} + \sin x \cdot e^{\sin x} \] Thus, the derivative of \( f(x) \) is \( f'(x) = \cos x e^{\sin x} + \sin x e^{\sin x} \), so the correct answer is (A).