Let's first compute the derivative of the function $f(x) = \frac{4 \sin x}{3 + \cos x}$.

Step 1: Apply the Quotient Rule

The quotient rule states that if we have a function of the form $\frac{g(x)}{h(x)}$, its derivative is:

$\left( \frac{g(x)}{h(x)} \right)' = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$

Here, $g(x) = 4 \sin x$ and $h(x) = 3 + \cos x$.

• Derivative of $g(x) = 4 \sin x$: $g'(x) = 4 \cos x$

• Derivative of $h(x) = 3 + \cos x$: $h'(x) = -\sin x$

Step 2: Apply the Derivatives

Now, we apply the quotient rule:

$f'(x) = \frac{(4 \cos x)(3 + \cos x) - (4 \sin x)(-\sin x)}{(3 + \cos x)^2}$

Simplify the numerator:

$f'(x) = \frac{4 \cos x (3 + \cos x) + 4 \sin^2 x}{(3 + \cos x)^2}$

Distribute $4 \cos x$ in the first term:

$f'(x) = \frac{12 \cos x + 4 \cos^2 x + 4 \sin^2 x}{(3 + \cos x)^2}$

Step 3: Use the Trigonometric Identity

Recall that $\sin^2 x + \cos^2 x = 1$, so:

$f'(x) = \frac{12 \cos x + 4 (1)}{(3 + \cos x)^2}$

Thus, the derivative simplifies to:

$f'(x) = \frac{12 \cos x + 4}{(3 + \cos x)^2}$

Step 4: Evaluate $f'(3)$

Now, we compute $f'(3)$.

First, find $\cos(3)$ and $\sin(3)$ (where 3 is in radians):

• $\cos(3) \approx -0.989992$
• $\sin(3) \approx 0.14112$

Substitute $x = 3$ into the formula for $f'(x)$:

$f'(3) = \frac{12(-0.989992) + 4}{(3 + (-0.989992))^2}$

$f'(3) = \frac{-11.879904 + 4}{(3 - 0.989992)^2}$

$f'(3) = \frac{-7.879904}{(2.010008)^2}$

$f'(3) = \frac{-7.879904}{4.04016}$

$f'(3) \approx -1.9507$

Final Answers:

• $f'(x) = \frac{12 \cos x + 4}{(3 + \cos x)^2}$
• $f'(3) \approx -1.9507$

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Do you want further details or have any questions about the steps?
Here are 5 relative questions for you:

1. How does the quotient rule work in other functions?
2. What are the key trigonometric identities needed in calculus?
3. How can you approximate trigonometric values for non-standard angles?
4. What happens to $f'(x)$ as $x$ approaches $\pi$?
5. Can you derive higher-order derivatives for the given function?

Tip: Always check for trigonometric identities when simplifying complex expressions involving sine and cosine.