let ( f(x)=3 x(sin (x)+cos (x)) ). find the following:\n1. ( f^{prime}(x)= )\n2. ( f^{prime}left(\frac{pi}{3}…

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})