Derivative At Zero 185Fe4

1. **State the problem:** We need to find the derivative of the function $f(x) = 2 \sin x + \sin^2 x$ at $x=0$. 2. **Recall the derivative rules:** - The derivative of $\sin x$ is $\cos x$. - Use the chain rule for $\sin^2 x = (\sin x)^2$. 3. **Find the derivative $f'(x)$:** $$f'(x) = \frac{d}{dx} \left(2 \sin x + \sin^2 x\right) = 2 \cos x + 2 \sin x \cos x$$ 4. **Evaluate $f'(0)$:** $$f'(0) = 2 \cos 0 + 2 \sin 0 \cos 0 = 2 \times 1 + 2 \times 0 \times 1 = 2$$ 5. **Final answer:** $$f'(0) = 2$$