let ( f(x)=-2 e^{x} sin x )
( f^{prime}(x)= )
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Accepted Answer

# Explanation:
## Step1: Apply the product rule
The product rule states that if $y = uv$, then $y^\prime=u^\prime v + uv^\prime$. Let $u=-2e^{x}$ and $v = \sin x$. First, find $u^\prime$ and $v^\prime$. The derivative of $u=-2e^{x}$ is $u^\prime=-2e^{x}$ (since $\frac{d}{dx}(e^{x})=e^{x}$), and the derivative of $v = \sin x$ is $v^\prime=\cos x$.
## Step2: Substitute into the product rule formula
$$
\begin{align*}
f^\prime(x)&=u^\prime v+uv^\prime\
&=- 2e^{x}\cdot\sin x+(-2e^{x})\cdot\cos x\
&=-2e^{x}(\sin x+\cos x)
\end{align*}
$$

# Answer:
$-2e^{x}(\sin x+\cos x)$

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Explanation:

Step1: Apply the product rule

Step2: Substitute into the product rule formula

Answer: