To differentiate the function

$y = \frac{-3x^2 - 6}{(x^2 - x - 6)^2},$

we'll use the quotient rule, which states that if $y = \frac{u}{v}$, then

$\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}.$

Here, let:

• $u = -3x^2 - 6$
• $v = (x^2 - x - 6)^2$

Step 1: Differentiate $u$ and $v$

1. Differentiate $u$: $\frac{du}{dx} = -6x.$

2. Differentiate $v$:

   • First, differentiate the inside: $w = x^2 - x - 6$ $\frac{dw}{dx} = 2x - 1.$
   • Now, use the chain rule for $v = w^2$: $\frac{dv}{dx} = 2w \cdot \frac{dw}{dx} = 2(x^2 - x - 6)(2x - 1).$

Step 2: Apply the Quotient Rule

Substituting into the quotient rule:

$\frac{dy}{dx} = \frac{(x^2 - x - 6)^2(-6x) - (-3x^2 - 6) \cdot 2(x^2 - x - 6)(2x - 1)}{(x^2 - x - 6)^4}.$

Step 3: Simplify

This expression can be simplified, but it can become quite complex. The derivative is:

$\frac{dy}{dx} = \frac{-6x(x^2 - x - 6)^2 + 2(-3x^2 - 6)(x^2 - x - 6)(2x - 1)}{(x^2 - x - 6)^4}.$

Final Expression

You can further simplify the numerator if needed, but the derivative is correctly expressed.

Would you like to see more steps in simplification, or do you have any specific questions? Here are some follow-up questions to consider:

1. Do you want to see the step-by-step simplification of the derivative?
2. Would you like to know about the applications of this differentiation in real-world problems?
3. Are you interested in how to find critical points using this derivative?
4. Do you want to explore the behavior of this function as $x$ approaches specific values?
5. Would you like help with related differentiation rules or techniques?

Tip: Always check for any common factors in the numerator and denominator to simplify before taking further steps.