We are given the function:

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

We need to find the derivative of $f(x)$. Let’s proceed step by step.

Step 1: Simplify the expression

First, simplify the constant part:

$f(x) = -2 \cdot \frac{1}{\sqrt{2x + 9}} = -2(2x + 9)^{-\frac{1}{2}}$

Step 2: Apply the chain rule

We can now differentiate using the chain rule. The general derivative of $(g(x))^n$ is:

$\frac{d}{dx} \left( g(x)^n \right) = n g(x)^{n-1} \cdot g'(x)$

Here, $g(x) = 2x + 9$ and $n = -\frac{1}{2}$.

Derivative of $2x + 9$:

$g'(x) = 2$

Now, apply the chain rule:

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

Step 3: Simplify the expression

Simplifying the terms:

$f'(x) = 2 \cdot (2x + 9)^{-\frac{3}{2}}$

Thus, the derivative of the given function is:

$f'(x) = 2(2x + 9)^{-\frac{3}{2}}$

Would you like more details on any step?

Related Questions:

1. How do you differentiate functions with square roots?
2. What is the chain rule, and why is it applied here?
3. Can you provide an example where the product rule is used along with the chain rule?
4. How do different rules for differentiation combine when dealing with complex functions?
5. How can the result be used to find tangent lines at specific points?

Tip:

When working with square roots or powers of functions, always check if constants can be simplified before differentiating to make the calculation easier!