if ( f(x)=3 x(sin x+cos x) ), find\n( f^{prime}(x)= )\n( f^{prime}(4)= )

Explanation:

Step1: Apply the product rule

The product rule states that if (y = u\cdot v), 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\ &=3(\sin x - x\sin x+\cos x+x\cos x)\ &=3\sin x(1 - x)+3\cos x(1 + x) \end{align*} ]

Step2: Substitute (x = 4) into (f^\prime(x))

[ \begin{align*} f^\prime(4)&=3\sin(4)(1 - 4)+3\cos(4)(1 + 4)\ &=-9\sin(4)+15\cos(4) \end{align*} ] Using a calculator (in radian mode), (\sin(4)\approx - 0.7568) and (\cos(4)\approx - 0.6536) [ \begin{align*} f^\prime(4)&=-9\times(-0.7568)+15\times(- 0.6536)\ &=6.8112-9.804\ &=-2.9928 \end{align*} ]

Answer:

(f^\prime(x)=3\sin x(1 - x)+3\cos x(1 + x)) (f^\prime(4)\approx - 2.99)