Derivative of -96x/(x^2+12)^2

1. The derivative of a constant times a function is the constant times the derivative of the function.

   1. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = x$ and $g{\left(x \right)} = \left(x^{2} + 12\right)^{2}$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = x^{2} + 12$.

      2. Apply the power rule: $u^{2}$ goes to $2 u$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 12\right)$:

         1. Differentiate $x^{2} + 12$ term by term:

            1. The derivative of the constant $12$ is zero.

            2. Apply the power rule: $x^{2}$ goes to $2 x$

            The result is: $2 x$

         The result of the chain rule is:

         $$2 x \left(2 x^{2} + 24\right)$$

      Now plug in to the quotient rule:

      $\frac{- 2 x^{2} \cdot \left(2 x^{2} + 24\right) + \left(x^{2} + 12\right)^{2}}{\left(x^{2} + 12\right)^{4}}$

   So, the result is: $- \frac{96 \left(- 2 x^{2} \cdot \left(2 x^{2} + 24\right) + \left(x^{2} + 12\right)^{2}\right)}{\left(x^{2} + 12\right)^{4}}$

2. Now simplify:

   $$\frac{288 \left(x^{2} - 4\right)}{\left(x^{2} + 12\right)^{3}}$$

The answer is:

$$\frac{288 \left(x^{2} - 4\right)}{\left(x^{2} + 12\right)^{3}}$$