Question

3.3b derivatives of trigonometric functions - ds2: problem 3
(6 points)
if
$f(x)=\frac{3\sin x}{2+\cos x}$
find $f(x)$.
answer:
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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, \(u = 3\sin x\), \(u^\prime=3\cos x\), \(v = 2+\cos x\), and \(v^\prime=-\sin x\).

Step2: Substitute into the quotient rule formula

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Step3: Use the trigonometric identity \(\sin^{2}x+\cos^{2}x = 1\)

Since \(\sin^{2}x+\cos^{2}x = 1\), then \(3\cos^{2}x+3\sin^{2}x=3(\cos^{2}x+\sin^{2}x)=3\).

$$ f^\prime(x)=\frac{6\cos x + 3}{(2+\cos x)^{2}}=\frac{3(2\cos x + 1)}{(2+\cos x)^{2}} $$

Answer:

\(\frac{3(2\cos x + 1)}{(2+\cos x)^{2}}\)