if ( f(x)=4x(sin x+cos x) ), find ( f^{prime}(x)= ) ( f^{prime}(2)= )

Answer

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 = 4x) and (v=\sin x+\cos x). Then (u^\prime = 4) and (v^\prime=\cos x-\sin x). [ \begin{align*} f^\prime(x)&=(4x)^\prime(\sin x+\cos x)+4x(\sin x+\cos x)^\prime\ &=4(\sin x+\cos x)+4x(\cos x - \sin x)\ &=4\sin x+4\cos x + 4x\cos x-4x\sin x \end{align*} ]

Step2: Evaluate (f^\prime(2))

Substitute (x = 2) into (f^\prime(x)): [ \begin{align*} f^\prime(2)&=4\sin(2)+4\cos(2)+4\times2\cos(2)-4\times2\sin(2)\ &=4\sin(2)+4\cos(2)+8\cos(2)-8\sin(2)\ &=(4\sin(2)-8\sin(2))+(4\cos(2)+8\cos(2))\ &=- 4\sin(2)+12\cos(2) \end{align*} ] Using a calculator ((\sin(2)\approx0.9093), (\cos(2)\approx - 0.4161)): [ \begin{align*} f^\prime(2)&=-4\times0.9093+12\times(-0.4161)\ &=-3.6372-4.9932\ &=-8.6304 \end{align*} ]

Answer:

(f^\prime(x)=4\sin x + 4\cos x+4x\cos x - 4x\sin x)

(f^\prime(2)\approx - 8.63)