https://file.aiquickdraw.com/mathbot/file/1727062533851lpurj02o.jpg

Let's analyze the problem shown in the image:

The function given is:

\[
f(x) = \frac{3\sqrt{x}}{6x^2 + 5x - 11}
\]

You are asked to compute the derivative \( f'(x) \) and evaluate it at \( x = 3 \), which gives:

\[
f'(3) = \frac{-47\sqrt{3}}{1682}
\]

To approach this, we can use the **quotient rule** for derivatives, since the function is a ratio of two functions. The quotient rule is:

\[
\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
\]

Where:
- \( u(x) = 3\sqrt{x} \)
- \( v(x) = 6x^2 + 5x - 11 \)

### Steps to Derive \( f'(x) \):

1. **Differentiate the numerator** \( u(x) = 3\sqrt{x} \) using the chain rule:
   \[
   u'(x) = 3 \cdot \frac{1}{2\sqrt{x}} = \frac{3}{2\sqrt{x}}
   \]

2. **Differentiate the denominator** \( v(x) = 6x^2 + 5x - 11 \):
   \[
   v'(x) = 12x + 5
   \]

3. Now apply the quotient rule:
   \[
   f'(x) = \frac{\left( \frac{3}{2\sqrt{x}} \right) \cdot (6x^2 + 5x - 11) - (3\sqrt{x}) \cdot (12x + 5)}{(6x^2 + 5x - 11)^2}
   \]

4. Finally, plug in \( x = 3 \) into the derivative to find \( f'(3) \).

The final result provided in the image is:

\[
f'(3) = \frac{-47\sqrt{3}}{1682}
\]

This suggests the computation simplifies to this expression after plugging in the values and simplifying.

Would you like to see the full step-by-step computation for plugging in \( x = 3 \), or do you have any other questions?

---

Here are five follow-up questions to deepen your understanding:
1. How does the chain rule apply when differentiating functions involving square roots?
2. What happens if you apply the product rule instead of the quotient rule in this scenario?
3. Can you explain why the denominator is squared in the quotient rule?
4. How would you find the second derivative \( f''(x) \) for the same function?
5. What other methods could be used to differentiate a ratio of functions involving roots?

**Tip:** When differentiating square root functions, always rewrite them as powers (e.g., \( \sqrt{x} = x^{1/2} \)) to make applying rules like the chain rule easier.

Evaluate the derivative f'(x) of the function f(x) = (3√x) / (6x^2 + 5x - 11) and compute f'(3).

Learn how to differentiate the function f(x) = (3√x) / (6x^2 + 5x - 11) using the quotient rule and chain rule. A detailed step-by-step solution is provided, including the computation of f'(3), resulting in -47√3/1682. Suitable for students in calculus.

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