if ( f(x)=\frac{3 sin x}{2+cos x} ), then ( f^{prime}(x)= ) ( f^{prime}(5)= )

Question

Accepted Answer

# Explanation:
## Step1: Apply the quotient rule
The quotient rule states that if $y = \frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 3\sin x$, $u^\prime=3\cos x$, $v = 2+\cos x$, $v^\prime=-\sin x$.
$$
\begin{align*}
f^\prime(x)&=\frac{(3\cos x)(2 + \cos x)-3\sin x(-\sin x)}{(2+\cos x)^{2}}\
&=\frac{6\cos x+3\cos^{2}x + 3\sin^{2}x}{(2+\cos x)^{2}}
\end{align*}
$$
## Step2: Use the trigonometric identity $\sin^{2}x+\cos^{2}x = 1$
Substitute $\sin^{2}x+\cos^{2}x = 1$ into the numerator.
$$
\begin{align*}
f^\prime(x)&=\frac{6\cos x+3(\cos^{2}x+\sin^{2}x)}{(2+\cos x)^{2}}\
&=\frac{6\cos x + 3\times1}{(2+\cos x)^{2}}\
&=\frac{6\cos x+3}{(2+\cos x)^{2}}
\end{align*}
$$
## Step3: Calculate $f^\prime(5)$
Substitute $x = 5$ into $f^\prime(x)$.
$$
f^\prime(5)=\frac{6\cos(5)+3}{(2+\cos(5))^{2}}
$$
Using a calculator (in radian mode), $\cos(5)\approx0.283662$
$$
\begin{align*}
f^\prime(5)&=\frac{6\times0.283662+3}{(2 + 0.283662)^{2}}\
&=\frac{1.701972+3}{(2.283662)^{2}}\
&=\frac{4.701972}{5.215077}\
&\approx0.9016
\end{align*}
$$

# Answer:
$f^\prime(x)=\frac{6\cos x + 3}{(2+\cos x)^{2}}$, $f^\prime(5)\approx0.9016$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply the quotient rule

Step2: Use the trigonometric identity \(\sin^{2}x+\cos^{2}x = 1\)

Step3: Calculate \(f^\prime(5)\)

Answer: