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

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

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the product rule

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

Answer: