Derivative of root^4(2x^2+6)^5

1. Let $u = \sqrt{2 x^{2} + 6}$.

2. Apply the power rule: $u^{1024}$ goes to $1024 u^{1023}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x^{2} + 6}$:

   1. Let $u = 2 x^{2} + 6$.

   2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{2} + 6\right)$:

      1. Differentiate $2 x^{2} + 6$ term by term:

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

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

            So, the result is: $4 x$

         2. The derivative of the constant $6$ is zero.

         The result is: $4 x$

      The result of the chain rule is:

      $$\frac{2 x}{\sqrt{2 x^{2} + 6}}$$

   The result of the chain rule is:

   $$2048 x \left(2 x^{2} + 6\right)^{511}$$

4. Now simplify:

   $$13729595320261219429963801598162786434538870600286610818788926918371086366795312104245119281322909109954592622782961716074243975999433287625148056582230114304 x \left(x^{2} + 3\right)^{511}$$

The answer is: