If f(x)=$$ \frac{4\sin x}{3+\cos x} $$, then
f'(x)f′(x) =   

f'(3)f′(3) = 


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.

If f(x) = $$ \frac{4\sin x}{3+\cos x} $$, then f'(x) = ?

f'(3) = ?

Learn how to compute the derivative of f(x) = 4sin(x)/(3 + cos(x)) using the quotient rule and evaluate f'(3). This step-by-step solution involves applying the quotient rule, simplifying trigonometric expressions, and calculating the derivative at a specific value.

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