let ( f(x)=3 x(sin (x)+cos (x)) ). find the following:
1. ( f^{prime}(x)= )
2. ( f^{prime}left(\frac{pi}{3}
ight)= )

Question

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 $u = 3x$ and $v=\sin(x)+\cos(x)$. Then $u^\prime = 3$ and $v^\prime=\cos(x)-\sin(x)$.
$$
\begin{align*}
f^\prime(x)&=3(\sin(x)+\cos(x))+3x(\cos(x)-\sin(x))\
&=3\sin(x)+3\cos(x)+3x\cos(x)-3x\sin(x)
\end{align*}
$$
## Step2: Substitute $x = \frac{\pi}{3}$ into $f^\prime(x)$
We know that $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, $\cos(\frac{\pi}{3})=\frac{1}{2}$
$$
\begin{align*}
f^\prime(\frac{\pi}{3})&=3\times\frac{\sqrt{3}}{2}+3\times\frac{1}{2}+3\times\frac{\pi}{3}\times\frac{1}{2}-3\times\frac{\pi}{3}\times\frac{\sqrt{3}}{2}\
&=\frac{3\sqrt{3}}{2}+\frac{3}{2}+\frac{\pi}{2}-\frac{\pi\sqrt{3}}{2}\
&=\frac{3 + 3\sqrt{3}+\pi-\pi\sqrt{3}}{2}
\end{align*}
$$

# Answer:
1. $3\sin(x)+3\cos(x)+3x\cos(x)-3x\sin(x)$
2. $\frac{3 + 3\sqrt{3}+\pi-\pi\sqrt{3}}{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the product rule

Step2: Substitute \(x = \frac{\pi}{3}\) into \(f^\prime(x)\)

Answer: