https://file.aiquickdraw.com/mathbot/file/1727293625914vufbm4ua.jpg

To find the derivative of the given function:

\[
f(x) = -\frac{12}{6\sqrt{2x+9}},
\]

we can break the problem into smaller steps.

### Step 1: Simplify the function
First, we simplify the function by dividing the constants:
\[
f(x) = -\frac{12}{6\sqrt{2x+9}} = -\frac{2}{\sqrt{2x+9}}.
\]

### Step 2: Use the chain rule
We now apply the chain rule to differentiate this function. The function is of the form:

\[
f(x) = -2(2x+9)^{-\frac{1}{2}}.
\]

We differentiate using the chain rule:

\[
f'(x) = -2 \cdot \left( -\frac{1}{2} \right)(2x+9)^{-\frac{3}{2}} \cdot (2).
\]

### Step 3: Simplify the derivative
Now, simplify the expression:

\[
f'(x) = -2 \cdot \left( -\frac{1}{2} \right)(2)(2x+9)^{-\frac{3}{2}} = \frac{2}{(2x+9)^{\frac{3}{2}}}.
\]

Thus, the derivative of the function is:

\[
f'(x) = \frac{2}{(2x+9)^{\frac{3}{2}}}.
\]

Would you like more details on any step or have questions?

### Relative questions:
1. How does the chain rule work in differentiation?
2. What are the steps to apply the power rule in derivatives?
3. How do you simplify expressions with radicals and powers?
4. Can you explain how to handle negative exponents when differentiating?
5. What happens when you differentiate functions inside square roots?

**Tip:** When differentiating functions with square roots, rewrite the square root as a fractional exponent before applying the power and chain rules.

Find the derivative of f(x) = -12 / (6 * sqrt(2x + 9)).

Learn how to differentiate f(x) = -12 / (6 * sqrt(2x + 9)) by applying the chain rule and power rule. Follow the detailed solution, simplifying expressions and applying differentiation techniques step by step. Suitable for students in Grades 11-12 or early college-level calculus.

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