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

Question

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

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Apply the product rule

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

Answer: